The Database of Integer Sequences, Part 97
Part of the On-Line Encyclopedia of Integer Sequences
This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.
For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )
Seis.html: Welcome
index.html: Lookup
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demo1.html: Demos
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WebCam.html: WebCam
Submit.html: Contribute new sequence or comment
eishelp1.html: Internal format
eishelp2.html: Beautified format
transforms.html: Transforms
Spuzzle.html: Puzzles
Shot.html: Hot
classic.html: Classics
ol.html: Superseeker
JIS/index.html: Journal of Integer Sequences
pages.html: More pages
Maintained by: N. J. A. Sloane (njas@research.att.com),
home page: www.research.att.com/~njas/
(start)
%I A024275
%S A024275 0,1,6,116,4816,342736,37272576,5748462656,1193465153536,
%T A024275 320935235205376,108513125117853696,45057835625375568896,
%U A024275 22540295648947188269056,13370560809273727964041216
%V A024275 0,1,-6,116,-4816,342736,-37272576,5748462656,-1193465153536,
%W A024275 320935235205376,-108513125117853696,45057835625375568896,
%X A024275 -22540295648947188269056,13370560809273727964041216
%N A024275 Expansion of ln(1+sinh(x)*sin(x))/2.
%t A024275 Log[ 1+Sinh[ x ]*Sin[ x ]]/2 (* Even Part *)
%Y A024275 A009359.
%Y A024275 Sequence in context: A003425 A052465 A113015 this_sequence A100070 A135869 A054957
%Y A024275 Adjacent sequences: A024272 A024273 A024274 this_sequence A024276 A024277 A024278
%K A024275 sign
%O A024275 0,3
%A A024275 R. H. Hardin (rhh(AT)cadence.com)
%E A024275 Extended with signs 03/97.
%I A100070
%S A100070 6,117,5632,515625,77262336,17230990189,5360119185408,2219048868131217,
%T A100070 1180000000000000000,783948341202404638821,636404158746280870281216,
%U A100070 619884903445287035295372217,713552333492738487958741450752
%N A100070 Number a(n) of forests with two components in the complete bipartite graph K_{n,n}.
%C A100070 This sequence (a(n)) appears to dominate the sequence (n^{2n-2}) of the number of spanning trees in K_{n,n} for n>1. This shows that the sequence of independent set numbers for the cycle matroid of K_{n,n} is not monotone increasing unlike the complete graph K_{n}.
%D A100070 N. Eaton, W. Kook and L. Thoma, Monotonicity for complete graphs, preprint, 2004.
%F A100070 a(n)=2(n^{2}-n))^{n-1}+(1/2!)\sum_{x, y\in [n-1]}b(n, x, y), where b(n, x, y)=binom{n}{x} binom{n}{y}x^{y-1}y^{x-1}(n-x)^{n-y-1}(n-y)^{n-x-1}
%e A100070 a(2)=6 because K_{2,2} is C_{4} the cycle of length 4, and there are 6 forests with two components in C_{4}.
%t A100070 a[n_]:=Sum[Binomial[n, x]*Binomial[n, y]*x^(y-1)*y^(x-1)*(n-x)^(n-y-1)*(n-y)^(n-x-1), {x, 1, n-1}, {y, 1, n-1}]/2 + (2*(n^2-n)^(n-1)); Table[a[n], {n, 2, 10}] (* This will generate a(n) from n=2 to 10. *)
%Y A100070 Cf. A069087, A083483, A000272.
%Y A100070 Sequence in context: A052465 A113015 A024275 this_sequence A135869 A054957 A081537
%Y A100070 Adjacent sequences: A100067 A100068 A100069 this_sequence A100071 A100072 A100073
%K A100070 nonn
%O A100070 2,1
%A A100070 Woong Kook (andrewk(AT)math.uri.edu), Nov 02 2004
%I A135869
%S A135869 1,1,6,117,6642,1097874,537135948,784812995973,3435153688724346,
%T A135869 45086429284345043334,1775007791598340247784372,
%U A135869 209630197234751724563143145346,74271350069687203431923556331222068
%N A135869 G.f. A(x) = 1 + x*A(3x)^2.
%C A135869 Self-convolution equals A135870.
%o A135869 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,3*x)^2);polcoeff(A,n)}
%Y A135869 Cf. A135867, A135870.
%Y A135869 Sequence in context: A113015 A024275 A100070 this_sequence A054957 A081537 A127726
%Y A135869 Adjacent sequences: A135866 A135867 A135868 this_sequence A135870 A135871 A135872
%K A135869 nonn
%O A135869 0,3
%A A135869 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 02 2007
%I A054957
%S A054957 1,1,6,118,7000,1329496,868255024,2039295163312,17639804273910144,
%T A054957 569596637165777524096,69273803156588266525129984
%N A054957 Number of labeled connected Eulerian digraphs with n nodes.
%H A054957 V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.
%Y A054957 Cf. A058338, A054955, A007080, A054959, A054956, A058337, A054958.
%Y A054957 Sequence in context: A024275 A100070 A135869 this_sequence A081537 A127726 A117063
%Y A054957 Adjacent sequences: A054954 A054955 A054956 this_sequence A054958 A054959 A054960
%K A054957 nonn,easy
%O A054957 1,3
%A A054957 njas, May 24 2000
%E A054957 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 06 2001
%I A081537
%S A081537 1,0,6,120,60
%N A081537 LCM of row n of triangle in A081536.
%Y A081537 Cf. A081535, A081536, A081538.
%Y A081537 Sequence in context: A100070 A135869 A054957 this_sequence A127726 A117063 A001219
%Y A081537 Adjacent sequences: A081534 A081535 A081536 this_sequence A081538 A081539 A081540
%K A081537 more,nonn
%O A081537 1,3
%A A081537 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 28 2003
%I A127726
%S A127726 6,120,126,2520,2640,30240,32640,37800,37926,55440,685440,758520,831600,
%T A127726 2600640,5533920,6917400,9102240,10281600,11377800,16687440,152182800,
%U A127726 206317440,250311600,475917120,866829600
%N A127726 Numbers n that are 3-imperfect.
%Y A127726 Cf. A127724 (k-imperfect numbers).
%Y A127726 Sequence in context: A135869 A054957 A081537 this_sequence A117063 A001219 A076233
%Y A127726 Adjacent sequences: A127723 A127724 A127725 this_sequence A127727 A127728 A127729
%K A127726 nonn
%O A127726 1,1
%A A127726 T. D. Noe (noe(AT)sspectra.com), Jan 25 2007
%I A117063
%S A117063 0,1,6,120,153,190,231,630,703,780,1035,1540,1770,2016,2701,2850,3003,
%T A117063 3160,4005,4560,4950,6670,6903,7140,9180,9730,10011,10296,10585,10878,
%U A117063 12090,12403,12720,13041,14028,14706,15051,15400,16110,17205,19110
%N A117063 Hexagonal numbers for which the product of the digits is also a hexagonal number.
%e A117063 24531 is in the sequence because (1) it is a hexagonal number and (2)the product of its digits 2*4*5*3*1=120 is also a hexagonal number.
%Y A117063 Cf. A000384.
%Y A117063 Sequence in context: A054957 A081537 A127726 this_sequence A001219 A076233 A066581
%Y A117063 Adjacent sequences: A117060 A117061 A117062 this_sequence A117064 A117065 A117066
%K A117063 base,nonn
%O A117063 0,3
%A A117063 Luc Stevens (lms022(AT)yahoo.com), Apr 16 2006
%I A001219
%S A001219 6,120,210,990,185136,258474216
%N A001219 Triangular numbers of form a(a+1)(a+2).
%D A001219 R. K. Guy, Unsolved Problems in Number Theory, D3.
%H A001219 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%Y A001219 Sequence in context: A081537 A127726 A117063 this_sequence A076233 A066581 A054776
%Y A001219 Adjacent sequences: A001216 A001217 A001218 this_sequence A001220 A001221 A001222
%K A001219 nonn,fini,full
%O A001219 1,1
%A A001219 njas
%I A076233
%S A076233 1,6,120,496,672,8128,30240,32760,523776,23569920,33550336,459818240,
%T A076233 1379454720,1476304896,8589869056,31998395520,51001180160,66433720320
%N A076233 Sigma[1, n]/n, Sigma[3, n]/n are integers.
%Y A076233 Cf. A007691, A046763, A055709, A076231, A076234.
%Y A076233 Sequence in context: A127726 A117063 A001219 this_sequence A066581 A054776 A076231
%Y A076233 Adjacent sequences: A076230 A076231 A076232 this_sequence A076234 A076235 A076236
%K A076233 nonn
%O A076233 1,2
%A A076233 Labos E. (labos(AT)ana.sote.hu), Oct 04 2002
%I A066581
%S A066581 1,6,120,504,2,60,336,288,40320,15552,1837080,1327104,309657600,
%T A066581 393750000,2015539200,94097687040,1366159011840,54793045278720,
%U A066581 140587147048320,720,3024,120960,10321920,28343520,334430208
%N A066581 Product of nonzero digits of A066547(n).
%e A066581 The third term of A066547 is 456 hence a(3) = 120.
%Y A066581 Cf. A066547.
%Y A066581 Sequence in context: A117063 A001219 A076233 this_sequence A054776 A076231 A076234
%Y A066581 Adjacent sequences: A066578 A066579 A066580 this_sequence A066582 A066583 A066584
%K A066581 easy,nonn,base
%O A066581 1,2
%A A066581 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 21 2001
%E A066581 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 01 2003
%I A054776
%S A054776 0,6,120,504,1320,2730,4896,7980,12144,17550,24360,32736,42840,54834,
%T A054776 68880,85140,103776,124950,148824,175560,205320,238266,274560,314364,
%U A054776 357840,405150,456456,511920,571704,635970,704880,778596,857280,941094
%N A054776 3n*(3n-1)*(3n-2).
%D A054776 Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 268
%D A054776 L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 46
%H A054776 Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
%F A054776 a(n)=A007531(3n-2)
%F A054776 sum(n=1, inf, 1/a(n))=Pi*sqrt(3)/12-ln(3)/4 - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002
%F A054776 G.f.: 6x(1+16x+10x^2)/(1-x)^4.
%F A054776 1/6 + 1/120 + 1/504 +...= (1/4)*(Pi/sqrt(3) - ln 3) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2006
%o A054776 (PARI) a(n)=3*n*(3*n-1)*(3*n-2)
%Y A054776 Sequence in context: A001219 A076233 A066581 this_sequence A076231 A076234 A066289
%Y A054776 Adjacent sequences: A054773 A054774 A054775 this_sequence A054777 A054778 A054779
%K A054776 easy,nonn
%O A054776 0,2
%A A054776 Henry Bottomley (se16(AT)btinternet.com), May 19 2000
%I A076231
%S A076231 1,6,120,672,8128,30240,32760,33550336,459818240,1379454720,1476304896,
%T A076231 8589869056,31998395520,51001180160,66433720320
%N A076231 Numbers n such that Sigma[1,n]/n, Sigma[3,n]/n and Sigma[5,n]/n are integers.
%Y A076231 Intersection of A007691, A046763 and A055709.
%Y A076231 Cf. A007691, A046763, A055709, A076233, A076234.
%Y A076231 Sequence in context: A076233 A066581 A054776 this_sequence A076234 A066289 A115678
%Y A076231 Adjacent sequences: A076228 A076229 A076230 this_sequence A076232 A076233 A076234
%K A076231 nonn
%O A076231 1,2
%A A076231 Labos E. (labos(AT)ana.sote.hu), Oct 03 2002
%I A076234
%S A076234 1,6,120,672,30240,32760,33550336,459818240,1379454720,8589869056,
%T A076234 31998395520,51001180160
%N A076234 Numbers n such that Sigma[1,n]/n, Sigma[3,n]/n, Sigma[5,n]/n and Sigma[7,n]/n are integers.
%Y A076234 Cf. A007691, A046763, A055709, A076231, A076233.
%Y A076234 Cf. A066289 (n divides sigma_k(n) for all odd k).
%Y A076234 Sequence in context: A066581 A054776 A076231 this_sequence A066289 A115678 A048604
%Y A076234 Adjacent sequences: A076231 A076232 A076233 this_sequence A076235 A076236 A076237
%K A076234 nonn
%O A076234 1,2
%A A076234 Labos E. (labos(AT)ana.sote.hu), Oct 04 2002
%I A066289
%S A066289 1,6,120,672,30240,32760,31998395520,796928461056000,212517062615531520,680489641226538823680000,
%T A066289 13297004660164711617331200000,1534736870451951230417633280000,
%U A066289 6070066569710805693016339910206758877366156437562171488352958895095808000000000
%N A066289 Numbers n such that Mod[DivisorSigma[2k-1,n],n]=0 holds for all k; i.e. all odd-power-sums of divisors of n are divisible by n.
%C A066289 Tested for each n and k<200. Otherwise the proof for all k seems laborious, since the number of divisors of terms of sequence rapidly increases: {1, 4, 16, 24, 96, 96, 2304, ...}.
%C A066289 Tested for each n and k<=1000. - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Oct 10 2003
%C A066289 The given terms have been tested for all k. - Don Reble, Nov 03, 2003
%C A066289 This is a proper subset of the multiply perfect numbers A007691. E.g. 8128 from A007691 is not here because its remainder at Sigma[odd,8128]/8128 division is 0 or 896 depending on odd exponent.
%F A066289 DivisorSigma[2k-1, n]/n is an integer for all k=1, 2, 3, .., 200, ...
%Y A066289 Cf. A066135, A066284, A007691, A066290.
%Y A066289 Sequence in context: A054776 A076231 A076234 this_sequence A115678 A048604 A001516
%Y A066289 Adjacent sequences: A066286 A066287 A066288 this_sequence A066290 A066291 A066292
%K A066289 nonn
%O A066289 1,2
%A A066289 Labos E. (labos(AT)ana.sote.hu), Dec 12 2001
%E A066289 The following numbers belong to the sequence, but there may be missing terms in between: 796928461056000 (also belongs to A046060); 212517062615531520 (also belongs to A046060); 680489641226538823680000 (also belongs to A046061); 13297004660164711617331200000 (also belongs to A046061) - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Oct 10 2003
%E A066289 Extended to 13 confirmed terms by Don Reble, Nov 04, 2003. There is a question whether there are other members below a[13]. However, there are none in Achim's list of multiperfect numbers (see A007691); Rich Schroppel has suggested that that list is complete to 10^70 - if so, a[1..12] are correct; as for a[13], Rich says there's only "an epsilon chance that some undiscovered MPFN lies in the gap." So it is very likely to be correct. - Don Reble
%I A115678
%S A115678 6,120,946,1540,7021,13366,14365,15400,19306,35245,72010,95266,156520,
%T A115678 179101,191890,334153,341551,913276,925480,933661,946000,1008910,
%U A115678 1030330,1131760,1383616,1945378,3066526,3156328,3308878,3584503
%N A115678 Triangular numbers whose digit reversal is a brilliant number (A078972).
%e A115678 946=T(43) and 649=11*59 is brilliant.
%Y A115678 Cf. A078972, A115677.
%Y A115678 Sequence in context: A076231 A076234 A066289 this_sequence A048604 A001516 A026337
%Y A115678 Adjacent sequences: A115675 A115676 A115677 this_sequence A115679 A115680 A115681
%K A115678 nonn,base
%O A115678 1,1
%A A115678 Giovanni Resta (g.resta(AT)iit.cnr.it), Jan 31 2006
%I A048604
%S A048604 1,6,120,1680,362880,7983360,6227020800,186810624000,355687428096000,
%T A048604 121645100408832000,51090942171709440000,213653030899875840000,
%U A048604 1723467782592331776000000,64431180179990249472000000
%N A048604 Denominators of coefficients in function a(x) such that a(a(x)) = arctan x.
%C A048604 Recursion exists for coefficients, but is too complicated to process without computer algebra system
%D A048604 W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
%D A048604 W. C. Yang, Composition equations, preprint, 1999
%D A048604 W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245.
%e A048604 x - x^3/6 + x^5 * 7/120 ...
%Y A048604 Cf. A048605.
%Y A048604 Sequence in context: A076234 A066289 A115678 this_sequence A001516 A026337 A065888
%Y A048604 Adjacent sequences: A048601 A048602 A048603 this_sequence A048605 A048606 A048607
%K A048604 frac,nonn
%O A048604 0,2
%A A048604 Winston C. Yang (yang(AT)math.wisc.edu)
%I A001516 M4295 N1795
%S A001516 0,0,6,120,1980,32970,584430,11204676,233098740,5254404210,127921380840,
%T A001516 3350718545460,94062457204716,2819367702529560,89912640142178490,
%U A001516 3040986592542420060,108752084073199561140,4101112025363285051526
%N A001516 Bessel polynomial {y_n}''(1).
%D A001516 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A001516 Index entries for sequences related to Bessel functions or polynomials
%Y A001516 Cf. A001518, A065944.
%Y A001516 Sequence in context: A066289 A115678 A048604 this_sequence A026337 A065888 A075844
%Y A001516 Adjacent sequences: A001513 A001514 A001515 this_sequence A001517 A001518 A001519
%K A001516 nonn,easy
%O A001516 0,3
%A A001516 njas
%I A026337
%S A026337 0,6,120,2016,32640,523776,8386560,134209536,2147450880,34359607296,
%T A026337 549755289600,8796090925056,140737479966720,2251799780130816,36028796884746240,
%U A026337 576460751766552576,9223372034707292160,147573952581086478336,2361183241400462868480
%N A026337 4^n*(4^n-1)/2.
%F A026337 a(n)=C(4^n,2),n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008
%p A026337 seq(binomial(4^n,2),n=0..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008
%Y A026337 Sequence in context: A115678 A048604 A001516 this_sequence A065888 A075844 A029697
%Y A026337 Adjacent sequences: A026334 A026335 A026336 this_sequence A026338 A026339 A026340
%K A026337 nonn
%O A026337 0,2
%A A026337 njas
%I A065888
%S A065888 6,120,2160,41160,860160,19840464,504000000,14030763120,425681879040,
%T A065888 13997939172360,496360987938816,18891066796875000,768426686420090880,
%U A065888 33279382190563948320,1529238539734890577920,74326797938267012471904
%N A065888 a(n) = number of endofunctions on [n] with a 4-cycle a->b->c->d->a, and for any x in [n], some iterate f^k(x) = a.
%F A065888 E.g.f.: T^4/4 where T = T(x) is Euler's tree function (see A000169).
%e A065888 a(4) = 6 : 3 [choices of 1's opposite in cycle] * 2 [choices of 1's image]
%Y A065888 Cf. A000169 (1-cycle), A053506 (2-cycle), A065513 (3-cycle), A065889 (= A065888/2: underlying simple graphs).
%Y A065888 Sequence in context: A048604 A001516 A026337 this_sequence A075844 A029697 A126448
%Y A065888 Adjacent sequences: A065885 A065886 A065887 this_sequence A065889 A065890 A065891
%K A065888 nonn
%O A065888 4,1
%A A065888 Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 27 2001
%I A075844
%S A075844 0,6,120,2394,47760,952806,19008360,379214394,7565279520,150926376006,
%T A075844 3010962240600,60068318435994,1198355406479280,23907039811149606,
%U A075844 476942440816512840,9514941776519107194,189821893089565631040
%N A075844 11*n^2 + 4 is a square.
%C A075844 Lim. n-> Inf. a(n)/a(n-1) = 10 + 3*Sqrt(11).
%D A075844 A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
%D A075844 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 341-400.
%D A075844 Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.
%H A075844 Tanya Khovanova, Recursive Sequences
%H A075844 J. J. O'Connor and E. F. Robertson, Pell's Equation
%H A075844 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%F A075844 a(n) = [(10+3*Sqrt(11))^n - (10-3*Sqrt(11))^n] / Sqrt(11); a(n) = 20*a(n-1) - a(n-2).
%F A075844 G.f.: 6x / (1 - 20x + x^2).
%Y A075844 Equals (1/3)[A075839(n+1)-A075839(n)].
%Y A075844 Sequence in context: A001516 A026337 A065888 this_sequence A029697 A126448 A126446
%Y A075844 Adjacent sequences: A075841 A075842 A075843 this_sequence A075845 A075846 A075847
%K A075844 nonn
%O A075844 0,2
%A A075844 Gregory V. Richardson (omomom(AT)hotmail.com), Oct 14 2002
%I A029697
%S A029697 6,120,3936,140160,5039616,181401600,6530359296,235092541440,
%T A029697 8463329918976,304679870791680,10968475323334656,394865111539384320,
%U A029697 14215144015015182336,511745184538935951360,18422826643395251798016
%N A029697 Number of words of length 2n in the 6 transpositions of S[ 4 ] equivalent to the identity.
%F A029697 (9*4^n+36^n)/12
%Y A029697 Sequence in context: A026337 A065888 A075844 this_sequence A126448 A126446 A057003
%Y A029697 Adjacent sequences: A029694 A029695 A029696 this_sequence A029698 A029699 A029700
%K A029697 nonn
%O A029697 1,1
%A A029697 Paolo Dominici (pl.dm(AT)libero.it)
%I A126448
%S A126448 1,6,120,4495,270725,24040016,2967205528,487444845680,103073959989495,
%T A126448 27319423696620550,8881600973913295056,3478625214672347911080,
%U A126448 1616770762998304775695925,880246034121663208464847200
%N A126448 Column 2 of triangle A126445; a(n) = C( C(n+4,3) - 4, n).
%o A126448 (PARI) a(n)=binomial((n+2)*(n+3)*(n+4)/3!-4, n)
%Y A126448 Cf. A126445; A126446, A126447, A126449.
%Y A126448 Sequence in context: A065888 A075844 A029697 this_sequence A126446 A057003 A096718
%Y A126448 Adjacent sequences: A126445 A126446 A126447 this_sequence A126449 A126450 A126451
%K A126448 nonn
%O A126448 0,2
%A A126448 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 27 2006
%I A126446
%S A126446 1,1,6,120,4845,324632,32468436,4529365776,840261910995,200063149171380,
%T A126446 59473554359599446,21592914273609648996,9403538945961296957821,
%U A126446 4838670732821812768919800,2904538537066424425438417800
%N A126446 Column 0 of triangle A126445; a(n) = C( C(n+2,3), n).
%o A126446 (PARI) a(n)=binomial(n*(n+1)*(n+2)/3!, n)
%Y A126446 Cf. A126445; A126447, A126448, A126449; A126451, A126455, A126458.
%Y A126446 Sequence in context: A075844 A029697 A126448 this_sequence A057003 A096718 A096720
%Y A126446 Adjacent sequences: A126443 A126444 A126445 this_sequence A126447 A126448 A126449
%K A126446 nonn
%O A126446 0,3
%A A126446 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 27 2006
%I A057003
%S A057003 1,6,120,5040,360360,39070080,5967561600,1220096908800,321570878428800,
%T A057003 106137499051584000,42873948150095462400,20803502274492921984000,
%U A057003 11938961126118491232768000,7998487694738166709923840000
%N A057003 Write the natural numbers in groups: 1; 2,3; 4,5,6; 7,8,9,10; ... and multiple the members of each group.
%C A057003 Each group begins with a triangular number + 1 and proceeds until the next triangular number.
%F A057003 (n (n + 1)/2)!/((n - 1) n /2)!
%t A057003 Table[(n (n + 1)/2)!/((n - 1) n /2)!, {n, 1, 15}]
%Y A057003 Cf. A006003.
%Y A057003 Sequence in context: A029697 A126448 A126446 this_sequence A096718 A096720 A009445
%Y A057003 Adjacent sequences: A057000 A057001 A057002 this_sequence A057004 A057005 A057006
%K A057003 nonn
%O A057003 0,2
%A A057003 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 09 2000
%I A096718
%S A096718 1,6,120,5040,362880,4435200,32947200,145297152000,69701632000,13516122267648000,
%T A096718 5676771352412160000,2872446304320552960000,1723467782592331776000000,1935799013407707050803200,
%U A096718 485144691014524112732160000,1353553687930522274522726400000,204193242064947360270857011200000
%N A096718 Denominators of terms in series expansion of arcsin(arctan(x)).
%e A096718 x-1/6*x^3+13/120*x^5-341/5040*x^7+18649/362880*x^9-177761/4435200*x^11+...
%Y A096718 Cf. A096717, A096664, A096671, A096712, A096716, A045688, A045689, A096721, A096722.
%Y A096718 Sequence in context: A126448 A126446 A057003 this_sequence A096720 A009445 A094273
%Y A096718 Adjacent sequences: A096715 A096716 A096717 this_sequence A096719 A096720 A096721
%K A096718 nonn,frac
%O A096718 0,2
%A A096718 njas, Aug 15 2004
%I A096720
%S A096720 1,6,120,5040,362880,13305600,2075673600,435891456000,13173608448000,13516122267648000,
%T A096720 5676771352412160000,2872446304320552960000,14243535393325056000000,241974876675963381350400000,
%U A096720 949196134593634133606400000,20303305318957834117840896000000,4288058083363894565687997235200000
%N A096720 Denominators of terms in series expansion of arctan(arcsin(x)).
%e A096720 x-1/6*x^3+13/120*x^5-173/5040*x^7+12409/362880*x^9-123379/13305600*x^11+...
%Y A096720 Cf. A096719, A096718, A096664, A096671, A096712, A096716, A045688, A045689.
%Y A096720 Sequence in context: A126446 A057003 A096718 this_sequence A009445 A094273 A094278
%Y A096720 Adjacent sequences: A096717 A096718 A096719 this_sequence A096721 A096722 A096723
%K A096720 nonn,frac
%O A096720 0,2
%A A096720 njas, Aug 15 2004
%I A009445
%S A009445 1,6,120,5040,362880,39916800,6227020800,1307674368000,355687428096000,
%T A009445 121645100408832000,51090942171709440000,25852016738884976640000,15511210043330985984000000,
%U A009445 10888869450418352160768000000,8841761993739701954543616000000,8222838654177922817725562880000000
%N A009445 (2n+1)!.
%C A009445 Denominators in the expansion of sin(x): sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...
%D A009445 W. Dunham, Touring the calculus gallery, Amer. Math. Monthly, 112 (2005), 1-19.
%D A009445 I. Newton, De analysi, 1669; reprinted in D. Whiteside, ed., The Mathematical Works of Isaac Newton, vol. 1, Johnson Reprint Co., 1964; see p. 20.
%D A009445 H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, p. 88.
%H A009445 Eric Weisstein's World of Mathematics, Hyperbolic Sine
%H A009445 Zerinvary Lajos, Sage Notebooks
%o A009445 sage: [stirling_number1(2*i,1) for i in xrange(1,22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008
%Y A009445 Cf. A010050, A000142.
%Y A009445 Sequence in context: A057003 A096718 A096720 this_sequence A094273 A094278 A093910
%Y A009445 Adjacent sequences: A009442 A009443 A009444 this_sequence A009446 A009447 A009448
%K A009445 nonn,easy
%O A009445 0,2
%A A009445 R. H. Hardin (rhh(AT)cadence.com), Joe Keane (jgk(AT)jgk.org)
%I A094273
%S A094273 1,6,120,5040,524160,14658134400,3055495622623660944000,
%T A094273 49836477033762340735543499750498132295731232000
%N A094273 Row products of triangle A094270.
%C A094273 For further terms see the b-file.
%H A094273 Martin Fuller, Table of n, a(n) for n = 1..11
%F A094273 a(n) = Prod_{k=1,..,n} A094270(n,k). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 23 2006
%e A094273 Product of the terms of the 4-th row = 7*8*9*10 = 5040. Product of the terms of the 5-th row = 12*13*14*15*16 = 524160 = 104*5040.
%Y A094273 Cf. A094270, A094271, A094272, A094274.
%Y A094273 Sequence in context: A096718 A096720 A009445 this_sequence A094278 A093910 A002370
%Y A094273 Adjacent sequences: A094270 A094271 A094272 this_sequence A094274 A094275 A094276
%K A094273 nonn
%O A094273 1,2
%A A094273 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 27 2004
%E A094273 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 23 2006
%E A094273 Edited by Martin Fuller (martin_n_fuller(AT)btinternet.com), Jun 13 2007
%I A094278
%S A094278 1,6,120,5040,720720,1396755360,1606268664000,17328426347232000,
%T A094278 26287222768750944000,5206394767133274466752000,
%U A094278 146523567931431743317801536000,170071077486556275922918785247488000
%N A094278 Product of terms in n-th row of triangle A094275.
%Y A094278 Cf. A094275.
%Y A094278 Sequence in context: A096720 A009445 A094273 this_sequence A093910 A002370 A012846
%Y A094278 Adjacent sequences: A094275 A094276 A094277 this_sequence A094279 A094280 A094281
%K A094278 nonn,easy,less
%O A094278 1,2
%A A094278 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 27 2004
%E A094278 Edited and extended by David Wasserman (dwasserm(AT)earthlink.net), Jul 26 2006
%I A093910
%S A093910 1,6,120,5040,5765760,19275223968000,13644281345408020027550269440000,
%T A093910 4402827357584746886229433170489943024971625310770489684257669120000000000
%N A093910 Group the natural numbers as in A093911; a(n) is the product of the n-th group.
%C A093910 Next term has 168 digits.
%F A093910 For n > 1, a(n) = prod_{i = A093911(n)..A093911(n+1)-1} i.
%Y A093910 Cf. A090904, A093911.
%Y A093910 Sequence in context: A009445 A094273 A094278 this_sequence A002370 A012846 A012641
%Y A093910 Adjacent sequences: A093907 A093908 A093909 this_sequence A093911 A093912 A093913
%K A093910 nonn,easy,less
%O A093910 1,2
%A A093910 Amarnath Murthy (amarnath_murthy(AT)yhaoo.com), Apr 24 2004
%E A093910 Edited and extended by David Wasserman (dwasserm(AT)earthlink.net), Mar 27 2006
%I A002370 M4296 N1796
%S A002370 1,1,6,120,5250,395010,45197460,7299452160,1580682203100,
%T A002370 441926274289500,154940341854097800,66565404923242024800
%N A002370 a(n)=(n-1)^2 a(n-2)-3C(n-1,3)a(n-4).
%D A002370 T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.
%D A002370 I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.
%D A002370 A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5.
%H A002370 T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 2.
%Y A002370 Sequence in context: A094273 A094278 A093910 this_sequence A012846 A012641 A012795
%Y A002370 Adjacent sequences: A002367 A002368 A002369 this_sequence A002371 A002372 A002373
%K A002370 nonn
%O A002370 0,3
%A A002370 njas
%I A012846
%S A012846 1,6,120,5376,415104,48143744,7784817664,1673986906112,
%T A012846 462531126525952,159859054841331712,67616604530628853760,
%U A012846 34373912421215807799296,20685291456673100857344000
%N A012846 sinh(sec(x)*arctanh(x))=x+6/3!*x^3+120/5!*x^5+5376/7!*x^7...
%Y A012846 Sequence in context: A094278 A093910 A002370 this_sequence A012641 A012795 A054479
%Y A012846 Adjacent sequences: A012843 A012844 A012845 this_sequence A012847 A012848 A012849
%K A012846 nonn
%O A012846 0,2
%A A012846 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A012641
%S A012641 1,6,120,6216,548544,74860544,14571230336,3827313412352,
%T A012641 1304171528695808,559299942314340352,294737591597024591872,
%U A012641 187195194238509459439616,141014910901971840963870720
%V A012641 1,-6,120,-6216,548544,-74860544,14571230336,-3827313412352,
%W A012641 1304171528695808,-559299942314340352,294737591597024591872,
%X A012641 -187195194238509459439616,141014910901971840963870720
%N A012641 tanh(arcsinh(x)*cos(x))=x-6/3!*x^3+120/5!*x^5-6216/7!*x^7...
%Y A012641 Sequence in context: A093910 A002370 A012846 this_sequence A012795 A054479 A012475
%Y A012641 Adjacent sequences: A012638 A012639 A012640 this_sequence A012642 A012643 A012644
%K A012641 sign
%O A012641 0,2
%A A012641 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A012795
%S A012795 1,6,120,6216,652800,115944576,31101732480,11717149191936,
%T A012795 5895158281052160,3816842537847607296,3090738173477544099840,
%U A012795 3060041827367678200774656,3636490261219371329053163520
%N A012795 arcsin(sec(x)*tan(x))=x+6/3!*x^3+120/5!*x^5+6216/7!*x^7+652800/9!*x^9...
%Y A012795 Sequence in context: A002370 A012846 A012641 this_sequence A054479 A012475 A053777
%Y A012795 Adjacent sequences: A012792 A012793 A012794 this_sequence A012796 A012797 A012798
%K A012795 nonn
%O A012795 0,2
%A A012795 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A054479
%S A054479 1,0,6,120,6300,514080,62785800,10676746080,2413521910800,
%T A054479 700039083744000,253445583029839200,112033456760809584000,
%U A054479 59382041886244720843200,37175286835046004765120000
%N A054479 Number of sets of cycle graphs of 2n nodes where the 2-colored edges alternate colors.
%C A054479 Also number of permutations in the symmetric group S_2n in which cycle lengths are even and greater than 2, cf. A130915. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 25 2007
%F A054479 If b(2n)=a(n) then egf of b is 1/(sqrt(e^x^2 * (1-x^2))).
%Y A054479 Cf. A001147, A001818, A053871.
%Y A054479 Sequence in context: A012846 A012641 A012795 this_sequence A012475 A053777 A023199
%Y A054479 Adjacent sequences: A054476 A054477 A054478 this_sequence A054480 A054481 A054482
%K A054479 nonn
%O A054479 0,3
%A A054479 Christian G. Bower (bowerc(AT)usa.net), Mar 29 2000
%I A012475
%S A012475 1,6,120,7056,758400,130918656,33173038080,11587651270656,
%T A012475 5337660866396160,3134856182239789056,2286367634020679024640,
%U A012475 2027369655249396662009856,2147914157998443571728875520
%V A012475 1,-6,120,-7056,758400,-130918656,33173038080,-11587651270656,
%W A012475 5337660866396160,-3134856182239789056,2286367634020679024640,
%X A012475 -2027369655249396662009856,2147914157998443571728875520
%N A012475 arctan(cos(x)*sin(x))=x-6/3!*x^3+120/5!*x^5-7056/7!*x^7+758400/9!*x^9...
%Y A012475 Sequence in context: A012641 A012795 A054479 this_sequence A053777 A023199 A007539
%Y A012475 Adjacent sequences: A012472 A012473 A012474 this_sequence A012476 A012477 A012478
%K A012475 sign
%O A012475 0,2
%A A012475 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A053777
%S A053777 1,6,120,10368,2582208,3143720448,11692182896640,219197554267521024,
%T A053777 12804488375721592356864,3325324798296500862330077184,
%U A053777 2537067900325971750395878897090560
%N A053777 Number of n X n binary matrices of order dividing 12 (i.e. number of solutions of X^12=I in GL(n,2)).
%D A053777 V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
%D A053777 Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%Y A053777 Cf. A053722, A053725, A053718.
%Y A053777 Sequence in context: A012795 A054479 A012475 this_sequence A023199 A007539 A040996
%Y A053777 Adjacent sequences: A053774 A053775 A053776 this_sequence A053778 A053779 A053780
%K A053777 nonn
%O A053777 1,2
%A A053777 Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 24 2000
%I A023199
%S A023199 1,6,120,27720,122522400,130429015516800,1970992304700453905270400,
%T A023199 1897544233056092162003806758651798777216000,
%U A023199 4368924363354820808981210203132513655327781713900627249499856876120704000
%N A023199 a(n) = least k with sigma(k) >= nk.
%C A023199 Following a suggestion from Ed Pegg Jr, the sequence can be written in a more readable form as: 1!, 3!, 5!, 11# * 3! * 2, 17# * 5! * 2, 29# * 7! * 4, 53# * 7! * 12, 89# * 11! * 2, 157# * 17# * 8! * 6, 271# * 23# * 10!, 487# * 29# * 10!, 857# * 37# * 11! * 42, 1487# * 53# * 15! * 2, ..., where p# = primorial(p) = A034386.
%C A023199 Comment from T. D. Noe (noe(AT)sspectra.com), Jul 06 2005:
%C A023199 "Let c(p) be the smallest colossally-abundant number having the prime factor p. See A073751 for info about computing these numbers.
%C A023199 Then the terms of this sequence can be expressed as
%C A023199 a(2) = c(3)
%C A023199 a(3) = c(5) * 2
%C A023199 a(4) = c(11) / 2
%C A023199 a(5) = c(17) / 3
%C A023199 a(6) = c(29) * 14
%C A023199 a(7) = c(53)
%C A023199 a(8) = c(89) * 4
%C A023199 a(9) = c(157) * 34
%C A023199 a(10) = c(271) * 23
%C A023199 a(11) = c(487) / 2
%C A023199 a(12) = c(857) / 2
%C A023199 a(13) = c(1487) * 212
%C A023199 a(14) = c(2621) * 710
%C A023199 a(15) = c(4561) * 506
%C A023199 a(16) = c(8011) / 2
%C A023199 a(17) = c(13999) * 1630"
%C A023199 Initially each term is divisible by the previous one. Is there a reason why this should always be true? - Santi Spadaro (santi_spadaro(AT)virgilio.it), Aug 13, 2002. The conjecture a(n)|a(n+1) holds out to n=10. - Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003. The conjecture a(n)|a(n+1) fails for n=15. - T. D. Noe (noe(AT)sspectra.com), Jul 08 2005.
%H A023199 T. D. Noe, An algorithm for finding the least k with sigma(k) >= nk
%Y A023199 A subsequence of A004394. The dominating primes are in A108402.
%Y A023199 Sequence in context: A054479 A012475 A053777 this_sequence A007539 A040996 A110442
%Y A023199 Adjacent sequences: A023196 A023197 A023198 this_sequence A023200 A023201 A023202
%K A023199 nonn
%O A023199 1,2
%A A023199 David W. Wilson (davidwwilson(AT)comcast.net)
%E A023199 More terms from wnissen(AT)tfn.net (Walter Nissen) Apr 15 1997. Further terms from Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003
%E A023199 The term a(10) = 271#23#10! was apparently found independently by Bodo Zinser and Don Reble, circa Jul 05 2005
%E A023199 The next term, a(11) = 487#29#10!, was corrected by Don Reble, Jul 06 2005
%E A023199 a(12) = 857#37#11!42 from Don Reble, Jul 06 2005
%E A023199 a(13) = 1487#53#15!2 found by T. D. Noe and confirmed by Don Reble, Jul 07 2005
%E A023199 a(14)-a(17) found by T. D. Noe and and rechecked by him Oct 11 2005
%I A007539 M4297
%S A007539 1,6,120,30240,14182439040,154345556085770649600,
%T A007539 141310897947438348259849402738485523264343544818565120000,
%U A007539 8268099687077761372899241948635962893501943883292455548843932421413884476391773708366277840568053624227289196057256213348352000000000
%N A007539 First n-fold perfect number.
%D A007539 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
%D A007539 A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 138.
%D A007539 R. K. Guy, Unsolved Problems in Number Theory, B2.
%H A007539 C. K. Caldwell, The Prime Glossary, multiply perfect
%H A007539 Achim Flammenkamp, The Multiply Perfect Numbers Page
%H A007539 Fred Helenius, Link to Glossary and Lists
%Y A007539 Cf. A000396, A005820, A027687, A046060, A046061.
%Y A007539 Cf. A007691, A072002.
%Y A007539 Sequence in context: A012475 A053777 A023199 this_sequence A040996 A110442 A137149
%Y A007539 Adjacent sequences: A007536 A007537 A007538 this_sequence A007540 A007541 A007542
%K A007539 nonn,nice
%O A007539 1,2
%A A007539 njas
%E A007539 More terms sent by Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 30 2000
%I A040996
%S A040996 1,6,120,32640,2147450880,9223372034707292160,
%T A040996 170141183460469231722463931679029329920,
%U A040996 57896044618658097711785492504343953926464851149359812787997104700240680714240
%N A040996 Maximum number of distinct functions at the bottom of a Boolean (or Binary) Decision Diagram (or BDD) with negation by pointer complementation.
%C A040996 At 0, the last variable, the only choice is (t, f) because the first entry is always uncomplemented and the 2nd must be different.
%C A040996 At level 1, the 2nd to last variable, the first entry is either t or a pointer to a following level (0), and the 2nd entry is either of these or its negation, except it may not equal the first entry.
%C A040996 At level n, the n-th to last variable, the first entry is either t or a pointer to one of the following levels' functions, and the second entry is any of these or its negation, but not equal to the first entry
%H A040996 David L. Dill, BDD's
%H A040996 Author?, More about BDD's
%F A040996 a(n) = (S(n-1) + 1) * (2*S(n-1) + 1) where S(n-1) = sum k= n.
%C A110442 See A110443 for the indices of these numbers in A004490. A073751 contains a program for quickly computing colossally abundant numbers.
%Y A110442 Cf. A004490 (colossally abundant numbers), A023199 (least number k such that sigma(k)/k >= n).
%Y A110442 Sequence in context: A023199 A007539 A040996 this_sequence A137149 A053710 A126244
%Y A110442 Adjacent sequences: A110439 A110440 A110441 this_sequence A110443 A110444 A110445
%K A110442 nonn
%O A110442 2,1
%A A110442 T. D. Noe (noe(AT)sspectra.com), Jul 20 2005
%I A137149
%S A137149 1,1,6,120,362880,39916800,1307674368000,355687428096000,
%T A137149 51090942171709440000,10888869450418352160768000000,
%U A137149 8841761993739701954543616000000
%N A137149 a(n)= (Prime[n])!/(Prime[n] EulerPhi[Prime[n]]).
%C A137149 Degree of Lagrange resolvent of polynomial prime degree. Ratio: degree of symmetric group of prime order n divided by order metacyclic group of prime order n. For degree of Lagrange resolvent of polynomial not prime degree see A137150.
%t A137149 Table[(Prime[n])!/(Prime[n] EulerPhi[Prime[n]]), {n, 1, 20}]
%Y A137149 Cf. A058161, A137150.
%Y A137149 Sequence in context: A007539 A040996 A110442 this_sequence A053710 A126244 A138572
%Y A137149 Adjacent sequences: A137146 A137147 A137148 this_sequence A137150 A137151 A137152
%K A137149 nonn
%O A137149 1,3
%A A137149 Artur Jasinski (grafix(AT)csl.pl), Jan 23 2008
%I A053710
%S A053710 6,120,3628880,51090942171709440000
%N A053710 Value of prime balanced factorials: n! is the mean of its 2 closest neighboring primes.
%F A053710 n! = (p+q)/2; p=n!+d, q=n!-d, p and q are the closest primes to n!
%e A053710 for n=21, n!=51090942171709440000, d=31 and the closest primes to 21! are q=21!-31=51090942171709439969, p=21!+31=51090942171709440031
%Y A053710 A033393, A033392, A006990, A037151, A006562, A053709.
%Y A053710 Sequence in context: A040996 A110442 A137149 this_sequence A126244 A138572 A078261
%Y A053710 Adjacent sequences: A053707 A053708 A053709 this_sequence A053711 A053712 A053713
%K A053710 nonn
%O A053710 1,1
%A A053710 Labos E. (labos(AT)ana.sote.hu), Feb 10 2000
%E A053710 The next two terms are 171! and 190! - Jud McCranie (j.mccranie(AT)comcast.net), Jul 04 2000
%I A126244
%S A126244 6,120,39916800,355687428096000,8841761993739701954543616000000,
%T A126244 33452526613163807108170062053440751665152000000000,
%U A126244 138683118545689835737939019720389406345902876772687432540821294940160000000000000
%N A126244 p! if p prime and p+2 prime.
%e A126244 3 and 5...........3!=6
%e A126244 5 and 7...........5!=120
%e A126244 17 and 19........17!=39916800
%p A126244 ZL:=[]:for p from 1 to 71 do if (isprime(p) and isprime(p+2)) then ZL:=[op(ZL),(p!)]; fi; od; print(ZL);
%Y A126244 Sequence in context: A110442 A137149 A053710 this_sequence A138572 A078261 A132507
%Y A126244 Adjacent sequences: A126241 A126242 A126243 this_sequence A126245 A126246 A126247
%K A126244 easy,nonn
%O A126244 1,1
%A A126244 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 08 2007
%I A138572
%S A138572 1,6,122,2126,7910,8254,16201,32312,32426,32998,65436
%N A138572 Numbers n such that n divides the sum of the digits of n^n in base 2.
%C A138572 The next term is larger than 100000.
%C A138572 Conjecture: the sequence is infinite.
%e A138572 6^6=1011011001000000; 1+0+1+1+0+1+1+0+0+1+0+0+0+0+0+0=6; (6 mod 6)=0
%Y A138572 Cf. A108827.
%Y A138572 Sequence in context: A137149 A053710 A126244 this_sequence A078261 A132507 A109820
%Y A138572 Adjacent sequences: A138569 A138570 A138571 this_sequence A138573 A138574 A138575
%K A138572 base,hard,more,nonn
%O A138572 1,2
%A A138572 Robert Gerbicz (robert.gerbicz(AT)gmail.com), May 12 2008
%I A078261
%S A078261 1,6,123,617,24681,6170253,1234050607,30851265177,12340506070809,
%T A078261 123405060708091,123405060708091011,1542563258851137639,
%U A078261 1234050607080910111213,61702530354045505560657
%N A078261 Smallest integer multiple of the decimal number N = 0.246...up to 2n (decimal point followed by concatenation of 2 through 2n of first n even numbers).
%Y A078261 Cf. A078260.
%Y A078261 Sequence in context: A053710 A126244 A138572 this_sequence A132507 A109820 A004993
%Y A078261 Adjacent sequences: A078258 A078259 A078260 this_sequence A078262 A078263 A078264
%K A078261 base,nonn
%O A078261 1,2
%A A078261 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 24 2002
%E A078261 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
%I A132507
%S A132507 6,123,186708,83166890053,8515561118862439596510,
%T A132507 89593611685468317268826980196874273126797827,
%U A132507 10209483208842759332361602803763687959277127003678203937063961691427797113039620625259938
%N A132507 Egyptian fraction representation for the cube root of 32.
%C A132507 Fractional part of 32^(1/3) = 1/6 + 1/123 + 1/186708 + ... Generated with Perl's Math::BigFloat module. Number of digits in terms is as follows: 1, 3, 6, 11, 22, 44, 89, 177, 353, ...
%e A132507 Cube root of 32 = 3.1748021039363989495034112785446165207829866557997...
%Y A132507 Sequence in context: A126244 A138572 A078261 this_sequence A109820 A004993 A133792
%Y A132507 Adjacent sequences: A132504 A132505 A132506 this_sequence A132508 A132509 A132510
%K A132507 frac,nonn
%O A132507 1,1
%A A132507 Jonathan Wellons (wellons(AT)gmail.com), Aug 23 2007
%I A109820
%S A109820 6,126,992,4921,18450
%N A109820 Column 9 of array illustrated in A089574 and related to A034261.
%e A109820 The associated sequences begin for n = 15 through 19:
%e A109820 ........................1.......5
%e A109820 ........................3.......18
%e A109820 ................3.......18......60
%e A109820 ........3.......18......60......150
%e A109820 1.......7.......25......65......140
%e A109820 ........................6.......42
%e A109820 ................12......84......324
%e A109820 ........12......84......324.....924
%e A109820 ........6.......42......162.....462
%e A109820 4.......32......132.....392.....952
%e A109820 ........................10......80
%e A109820 ................30......240.....1050
%e A109820 ........10......90......420.....1400
%e A109820 ........30......240.....1050....3360
%e A109820 1.......11......56......196.....546
%e A109820 ........................15......135
%e A109820 ................60......540.....2640
%e A109820 ........15......165.....900.....3420
%e A109820 ........................21......210
%e A109820 ................35......385.....2205
%e A109820 ........................28......308
%e A109820 ........................1.......19
%e A109820 therefore A109820 begins
%e A109820 6 126 992 4921 18450
%Y A109820 Cf. A109126.
%Y A109820 Sequence in context: A138572 A078261 A132507 this_sequence A004993 A133792 A081623
%Y A109820 Adjacent sequences: A109817 A109818 A109819 this_sequence A109821 A109822 A109823
%K A109820 easy,more,nonn
%O A109820 0,1
%A A109820 Alford Arnold (Alford1940(AT)aol.com), Jul 03 2005
%I A004993
%S A004993 1,6,126,3276,93366,2800980,86830380,2753763480,88808872230,
%T A004993 2901089826180,95735964263940,3185396629145640,106710787076378940,
%U A004993 3595332672265690440,121727691903852662040,4138741524730990509360
%N A004993 (6^n/n!)*product[ k=0..n-1 ](6*k + 1).
%F A004993 G.f.: A(x) = (1 - 36*x)^(-1/6).
%F A004993 a(n) ~ Gamma(1/6)^-1*n^(-5/6)*6^(2*n)*{1 - 5/72*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
%Y A004993 Sequence in context: A078261 A132507 A109820 this_sequence A133792 A081623 A089314
%Y A004993 Adjacent sequences: A004990 A004991 A004992 this_sequence A004994 A004995 A004996
%K A004993 nonn
%O A004993 0,2
%A A004993 Joe Keane (jgk(AT)jgk.org)
%E A004993 Corrected by Franklin T. Adams-Watters, Oct 25 2006
%I A133792
%S A133792 6,126,8950,2308238,2129167114,7077040829290,84651408281226926,
%T A133792 3644360411354234096546,564689913941224929404667818,
%U A133792 314919227342521055797569563796454
%N A133792 Number of n X n binary matrices with every 1 adjacent to some zero, and every 0 adjacent to some one, horizontally or vertically.
%Y A133792 Sequence in context: A132507 A109820 A004993 this_sequence A081623 A089314 A111873
%Y A133792 Adjacent sequences: A133789 A133790 A133791 this_sequence A133793 A133794 A133795
%K A133792 nonn
%O A133792 2,1
%A A133792 Ron Hardin (rhh(AT)cadence.com), Jan 05 2008
%I A081623
%S A081623 1,6,126,12870,5200300,9075135300,63205303218876,1832624140942590534,
%T A081623 212392290424395860814420,100891344545564193334812497256,
%U A081623 191645966716130525165099506263706416,1480212998448786189993816895482588794876100
%N A081623 Number of ways in which the points on an n X n square lattice can be equally occupied with spin "up" and spin "down" particles. If n is odd, we arbitrarily take the lattice to contain one more spin "up" particle than the number of spin "down" particles.
%D A081623 Brian Hayes, The World in a Spin, American Scientist, vol. 88, pp. 384-388 (2000).
%H A081623 Brian Hayes, The World in a Spin.
%F A081623 a(n) = C(n^2, (n^2+1)/2) if n is odd and C(n^2, n^2/2) if n is even
%e A081623 a(2)=6 because C(4,2)=6
%e A081623 a(3)=126 because C(9,5)=126
%o A081623 (Mathcad or Microsoft Excel): f(n)=combin(n^2,trunc((n^2+1)/2))
%Y A081623 Sequence in context: A109820 A004993 A133792 this_sequence A089314 A111873 A012842
%Y A081623 Adjacent sequences: A081620 A081621 A081622 this_sequence A081624 A081625 A081626
%K A081623 easy,nonn
%O A081623 1,2
%A A081623 Tim Royappa (royappa(AT)uwf.edu), Apr 22 2003
%I A089314
%S A089314 0,6,128,2220,32112,421004,5209896,62098788,720987680,8209876572,
%T A089314 92098765464,1020987654356,11209876543248,122098765432140,
%U A089314 1320987654321032,14209876543209924,152098765432098816
%N A089314 Sum of all digits in all even numbers from 0 to 444...4 (with n 4's).
%e A089314 a(2) = 0+2+4+6+8+1+0+1+2+1+4+..+4+4=128.
%Y A089314 Cf. A089304.
%Y A089314 Sequence in context: A004993 A133792 A081623 this_sequence A111873 A012842 A012638
%Y A089314 Adjacent sequences: A089311 A089312 A089313 this_sequence A089315 A089316 A089317
%K A089314 nonn
%O A089314 0,2
%A A089314 Yalcin Aktar (aktaryalcin(AT)msn.com), Dec 25 2003
%E A089314 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Sep 09 2005
%I A111873
%S A111873 0,6,128,2500,51840,1176490,29360128,803538792,24000000000,778122738030,
%T A111873 27243640258560,1025115745389164,41273168209215488,1771037512207031250,
%U A111873 80704505322479288320,3892895350053349478480,198189314749641818898432
%N A111873 The work performed by a partial function f:{1,...,n}->{1,...,n} is defined to be work(f)=sum(|i-f(i)|,i in dom(f)); a(n) is equal to sum(work(f)) where the sum is over all partial functions f:{1,...,n}->{1,...,n}.
%C A111873 If n == -1 (mod 10^k) then 10^(n*k) divides a(n), so 10^9 divides a(9), 10^19 divides a(19),...,10^198 divides a(99), etc. - Farideh Firoozbakht (f.firoozbakht(AT)math.ui.ac.ir), Nov 27 2005
%H A111873 James East The Work Performed by a Transformation Semigroup, preprint 2005.
%F A111873 (n+1)^n*(n^2-n)/3
%e A111873 When n=2 there are 9 partial maps {1,2}->{1,2}: these are (1 1), (2 2), (1 2), (2 1), (1 -), (2 -), (- 1), (- 2) (- -). Adding up the work performed by these maps (from left to right as arranged above) gives a(2)=1+1+0+2+0+1+1+0+0=6.
%t A111873 Table[(n + 1)^n*(n^2 - n)/3, {n, 17}] (* Robert G. Wilson v *)
%Y A111873 Cf. A111867, A111874, A111903.
%Y A111873 Sequence in context: A133792 A081623 A089314 this_sequence A012842 A012638 A095695
%Y A111873 Adjacent sequences: A111870 A111871 A111872 this_sequence A111874 A111875 A111876
%K A111873 easy,nonn
%O A111873 1,2
%A A111873 James East (jameseastseq(AT)hotmail.com), Nov 23 2005
%E A111873 More terms from Farideh Firoozbakht (f.firoozbakht(AT)math.ui.ac.ir) and Robert G. Wilson v (rgwv(at)rgwv.com), Nov 27 2005
%I A012842
%S A012842 1,6,128,7000,757760,137329280,37511412352,14396574922496,
%T A012842 7385535722319872,4879586077682790400,4034467892180241121280,
%U A012842 4080124101723746610511872,4954185596977233058289319936
%N A012842 arcsin(sec(x)*arctanh(x))=x+6/3!*x^3+128/5!*x^5+7000/7!*x^7...
%Y A012842 Sequence in context: A081623 A089314 A111873 this_sequence A012638 A095695 A000907
%Y A012842 Adjacent sequences: A012839 A012840 A012841 this_sequence A012843 A012844 A012845
%K A012842 nonn
%O A012842 0,2
%A A012842 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A012638
%S A012638 1,6,128,7784,863168,153777536,40156766336,14452100967680,
%T A012638 6857570007949312,4148412279035834368,3116288727548544729088,
%U A012638 2846053921629037944078336,3105570670613041972460814336
%V A012638 1,-6,128,-7784,863168,-153777536,40156766336,-14452100967680,
%W A012638 6857570007949312,-4148412279035834368,3116288727548544729088,
%X A012638 -2846053921629037944078336,3105570670613041972460814336
%N A012638 arctan(arcsinh(x)*cos(x))=x-6/3!*x^3+128/5!*x^5-7784/7!*x^7...
%Y A012638 Sequence in context: A089314 A111873 A012842 this_sequence A095695 A000907 A077031
%Y A012638 Adjacent sequences: A012635 A012636 A012637 this_sequence A012639 A012640 A012641
%K A012638 sign
%O A012638 0,2
%A A012638 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A095695
%S A095695 6,130,1005,4830,17465,52101,135135,314985,674685,1349205,2548546,
%T A095695 4587765,7925190,13210190,21341970,33540966,51434520,77158620,113477595,
%U A095695 163923760,232959111,326161275,450436025
%N A095695 T(n,4) diagonal of triangle in A095693.
%D A095695 Horne, Nicholas S. "Analysis of Viable Network Configurations from a Combinatorial, Graphical, and Algebraic Perspective." Diss. Providence College, 2004.
%F A095695 a(n) = ((n)(n-1)(n-2)(n-3)(n^4+2n^3-13n^2-54n+136)/384
%Y A095695 Sequence in context: A111873 A012842 A012638 this_sequence A000907 A077031 A137038
%Y A095695 Adjacent sequences: A095692 A095693 A095694 this_sequence A095696 A095697 A095698
%K A095695 easy,nonn
%O A095695 4,1
%A A095695 Nicholas S. Horne (nickhorne(AT)cox.net), Jul 06 2004
%I A000907 M4298 N1797
%S A000907 6,130,2380,44100,866250,18288270,416215800,10199989800,268438920750,
%T A000907 7562120816250,227266937597700,7262844156067500,246045975136211250,
%U A000907 8810836639999143750,332624558868351750000,13205706717164131170000
%N A000907 Second order reciprocal Stirling number (Fekete) [[2n+2 \over n]]. The number of n-orbit permutations of a (2n+2)-set with at least 2 elements in each orbit. Also known as associated Stirling numbers of the first kind (e.g. Comtet).
%D A000907 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
%D A000907 A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.
%D A000907 C. Jordan, On Stirling's Numbers, Tohoku Math. J., 37 (1933), 254-278.
%D A000907 C. Jordan, Calculus of Finite Differences. Budapest, 1939, p. 152.
%F A000907 [[2n+2, n]]=sum((-1)^i*binomial(2n+2, 2n+2-i)[2n+2-i, n-i] where [n, k] is the unsigned Stirling number of the first kind.
%p A000907 s1 := (n,k)->sum((-1)^i*binomial(n,i)*abs(stirling1(n-i,k-i)),i=0..n); for j from 1 to 20 do s1(2*j+2,j); od;
%Y A000907 Cf. A000483, A001784, A001785.
%Y A000907 Sequence in context: A012842 A012638 A095695 this_sequence A077031 A137038 A024276
%Y A000907 Adjacent sequences: A000904 A000905 A000906 this_sequence A000908 A000909 A000910
%K A000907 nonn
%O A000907 0,1
%A A000907 njas
%E A000907 More terms, Maple program, formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
%I A077031
%S A077031 6,130,5027,5028
%N A077031 Numbers n such that A068340(n)=+/-3.
%Y A077031 Cf. A068340, A076542, A077030-A077033.
%Y A077031 Sequence in context: A012638 A095695 A000907 this_sequence A137038 A024276 A137037
%Y A077031 Adjacent sequences: A077028 A077029 A077030 this_sequence A077032 A077033 A077034
%K A077031 easy,nonn
%O A077031 1,1
%A A077031 Zak Seidov (zakseidov(AT)yahoo.com), Oct 21 2002
%I A137038
%S A137038 1,6,131,876,881,8131,8876,11731,13181,18376,18381,37131,88776,116881,117181,117776,133181,183776,
%T A137038 336731,888131,1366881,1373376,3336731,3371376,3713181,6113376,7866881,11761731,13167131,13717776,
%U A137038 13718131,13738181,33336731,33367181,37118376,37133181,61383131,78668776,87367681,87616881,117383181
%N A137038 Numbers n such that n and the square of n use only the digits 1, 3, 6, 7 and 8.
%C A137038 Generated with DrScheme
%H A137038 Jonathan Wellons, Table of n, a(n) for n=1..179
%H A137038 J. Wellons, Tables of Shared Digits
%e A137038 878161817313181^2 = 771168177386788681117836338761
%Y A137038 Sequence in context: A095695 A000907 A077031 this_sequence A024276 A137037 A101131
%Y A137038 Adjacent sequences: A137035 A137036 A137037 this_sequence A137039 A137040 A137041
%K A137038 base,nonn
%O A137038 1,2
%A A137038 Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008
%I A024276
%S A024276 0,1,6,131,6636,478981,46186866,5850470471,949119838296,
%T A024276 191928333160201,47247581207762526,13887944994837826571,
%U A024276 4798123437751335493956,1923214499293440219360781
%N A024276 Expansion of sinh(tan(x)*sinh(x))/2.
%t A024276 Sinh[ Tan[ x ]*Sinh[ x ]]/2 (* Even Part *)
%Y A024276 A009610.
%Y A024276 Sequence in context: A000907 A077031 A137038 this_sequence A137037 A101131 A009688
%Y A024276 Adjacent sequences: A024273 A024274 A024275 this_sequence A024277 A024278 A024279
%K A024276 nonn
%O A024276 0,3
%A A024276 R. H. Hardin (rhh(AT)cadence.com)
%E A024276 Extended and signs tested 03/97.
%I A137037
%S A137037 1,6,131,11731,3336731,3371376,6113376,33336731,611361731
%N A137037 Numbers n such that n and the square of n use only the digits 1, 3, 6 and 7.
%C A137037 Generated with DrScheme
%H A137037 J. Wellons, Tables of Shared Digits
%e A137037 611361731^2 = 373763166131316361
%Y A137037 Sequence in context: A077031 A137038 A024276 this_sequence A101131 A009688 A132872
%Y A137037 Adjacent sequences: A137034 A137035 A137036 this_sequence A137038 A137039 A137040
%K A137037 base,nonn
%O A137037 1,2
%A A137037 Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008
%I A101131
%S A101131 0,6,132,937
%N A101131 Indices of primes in sequence defined by A(0) = 73, A(n) = 10*A(n-1) - 17 for n > 0.
%C A101131 Numbers n such that (640*10^n + 17)/9 is prime.
%C A101131 Numbers n such that digit 7 followed by n >= 0 occurrences of digit 1 followed by digit 3 is prime.
%C A101131 Numbers corresponding to terms <= 937 are certified primes.
%D A101131 Klaus Brockhaus and Walter Oberschelp, Zahlenfolgen mit homogenem Ziffernkern, MNU 59/8 (2006), pp. 462-467.
%e A101131 71111113 is prime, hence 6 is a term.
%o A101131 (PARI) a=73;for(n=0,1000,if(isprime(a),print1(n,","));a=10*a-17)
%o A101131 (PARI) for(n=0,1000,if(isprime((640*10^n+17)/9),print1(n,",")))
%Y A101131 Cf. A000533, A002275.
%Y A101131 a(n) = A103050(n) - 1.
%Y A101131 Sequence in context: A137038 A024276 A137037 this_sequence A009688 A132872 A015503
%Y A101131 Adjacent sequences: A101128 A101129 A101130 this_sequence A101132 A101133 A101134
%K A101131 nonn,hard,more
%O A101131 1,2
%A A101131 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de) and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 03 2004
%I A009688
%S A009688 1,6,132,6888,631120,88901472,17815778112,4813349520000,
%T A009688 1685527674636544,742370072201602560,401601528167832511488,
%U A009688 261760691714482869528576,202316143535276408972627968
%N A009688 Expansion of tan(sinh(x)/cos(x)).
%t A009688 Tan[ Sinh[ x ]/Cos[ x ]] (* Odd Part *)
%Y A009688 Sequence in context: A024276 A137037 A101131 this_sequence A132872 A015503 A003373
%Y A009688 Adjacent sequences: A009685 A009686 A009687 this_sequence A009689 A009690 A009691
%K A009688 nonn
%O A009688 0,2
%A A009688 R. H. Hardin (rhh(AT)cadence.com)
%E A009688 Extended and signs tested Mar 15 1997 by Olivier Gerard.
%I A132872
%S A132872 1,1,6,132,7156,729895,119636226,28619359629,9374688646296,
%T A132872 4019108763468573,2180474045020534600,1458451073246597456521,
%U A132872 1177921104348705716833164,1129393220849450436646366223
%N A132872 Column 0 of triangle A132870.
%C A132872 Triangle T=A132870 obeys: the g.f. of row n of T^n = (y + n^2)^n for n>=0.
%o A132872 (PARI) {a(n)=local(M=Mat(1),N,L);for(i=1,n,N=M; M=matrix(#N+1,#N+1,r,c,if(r>=c,if(r<=#N,(N^(#N))[r,c], polcoeff((x+(#M)^2)^(#M),c-1)))); L=sum(i=1,#M,-(M^0-M)^i/i);M=sum(i=0,#M,(L/#N)^i/i!);); M[n+1,1]}
%Y A132872 Cf. A132870, A132871, A132873.
%Y A132872 Sequence in context: A137037 A101131 A009688 this_sequence A015503 A003373 A129047
%Y A132872 Adjacent sequences: A132869 A132870 A132871 this_sequence A132873 A132874 A132875
%K A132872 nonn
%O A132872 0,3
%A A132872 Paul D. Hanna (pauldhanna(AT)juno.com), Sep 29 2007
%I A015503
%S A015503 1,1,6,132,11352,3882384,5303336544,28966824203328,632809241545903488,
%T A015503 55296137144764138588416,19327437631660830304254690816,
%U A015503 27021729207700270170039091739231232
%N A015503 a(1)=1, a(n) = sum_{k=1}^{k=n-1} (4^k-1)/3 a(k).
%Y A015503 Sequence in context: A101131 A009688 A132872 this_sequence A003373 A129047 A050281
%Y A015503 Adjacent sequences: A015500 A015501 A015502 this_sequence A015504 A015505 A015506
%K A015503 nonn,easy
%O A015503 1,3
%A A015503 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A003373
%S A003373 6,133,260,387,514,641,768,2192,2319,2446,2573,2700,2827,4378,4505,4632,
%T A003373 4759,4886,6564,6691,6818,6945,8750,8877,9004,10936,11063,13122,16389,
%U A003373 16516,16643,16770,16897,17024,18575,18702,18829,18956,19083,20761,20888
%N A003373 Numbers that are the sum of 6 positive 7-th powers.
%Y A003373 Sequence in context: A009688 A132872 A015503 this_sequence A129047 A050281 A096756
%Y A003373 Adjacent sequences: A003370 A003371 A003372 this_sequence A003374 A003375 A003376
%K A003373 nonn
%O A003373 1,1
%A A003373 njas
%I A129047
%S A129047 0,6,133,1971,23541,245261,2326207,20685641,175544819,1438903377,
%T A129047 11485167375
%N A129047 Number of n-node triangulations of the Klein bottle N_2 in which every node has degree >= 4.
%D A129047 G. Ringel, Wie man die geschlossenen nichtorientierbaren Flaechen in moeglichst wenig Dreiecke zerlegen kann, Math. Ann. 130 (1955), 317-326.
%H A129047 Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).
%Y A129047 Sequence in context: A132872 A015503 A003373 this_sequence A050281 A096756 A013299
%Y A129047 Adjacent sequences: A129044 A129045 A129046 this_sequence A129048 A129049 A129050
%K A129047 nonn
%O A129047 7,2
%A A129047 njas, May 13 2007
%I A050281
%S A050281 6,135,1735,4902,65260,963024,82599811,175820910,1270311937
%N A050281 a(n) is the starting position of the first occurrence of a string of n 2's in the decimal expansion of Pi.
%H A050281 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%Y A050281 Cf. A035117.
%Y A050281 Sequence in context: A015503 A003373 A129047 this_sequence A096756 A013299 A013295
%Y A050281 Adjacent sequences: A050278 A050279 A050280 this_sequence A050282 A050283 A050284
%K A050281 nonn,base
%O A050281 1,1
%A A050281 Eric Weisstein (eric(AT)weisstein.com)
%E A050281 More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
%I A096756
%S A096756 6,135,1735,4902,65260,963024,82599811,175820910,1270311937
%N A096756 Index of first occurrence of just n consecutive twos in a row in the decimal expansion of Pi.
%C A096756 Presently identical to A050281.
%H A096756 David G. Andersen, The Pi-Search Page.
%Y A096756 Cf. A050281, A035117, A096757, A096758, A096759, A096760, A096761, A096762, A096763, A050279.
%Y A096756 Sequence in context: A003373 A129047 A050281 this_sequence A013299 A013295 A090407
%Y A096756 Adjacent sequences: A096753 A096754 A096755 this_sequence A096757 A096758 A096759
%K A096756 base,nonn
%O A096756 1,1
%A A096756 Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 07 2004
%I A013299
%S A013299 1,6,135,6300,496125,58939650,9833098275,2191376187000,
%T A013299 628651043645625,225615874552818750,99022807341232149375,
%U A013299 52176017395434685252500,32501310835906189355203125
%N A013299 sinh(log(x+1)-arctanh(x))=-1/2!*x^2-6/4!*x^4-135/6!*x^6-6300/8!*x^8...
%C A013299 Number of degree-2n permutations without odd cycles and with odd number of even cycles, offset 1. E.g.f.: x^2/(2*sqrt(1-x^2)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 10 2007
%Y A013299 Cf. A013302.
%Y A013299 Sequence in context: A129047 A050281 A096756 this_sequence A013295 A090407 A075185
%Y A013299 Adjacent sequences: A013296 A013297 A013298 this_sequence A013300 A013301 A013302
%K A013299 nonn
%O A013299 0,2
%A A013299 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A013295
%S A013295 1,6,135,6300,503685,61434450,10620124515,2471073204600,
%T A013295 744907219281225,282494651079390750,131664032748125452575,
%U A013295 73997676994079578439700,49364071776588936300058125
%N A013295 arcsin(log(x+1)-arctanh(x))=-1/2!*x^2-6/4!*x^4-135/6!*x^6-6300/8!*x^8...
%Y A013295 Sequence in context: A050281 A096756 A013299 this_sequence A090407 A075185 A003994
%Y A013295 Adjacent sequences: A013292 A013293 A013294 this_sequence A013296 A013297 A013298
%K A013295 nonn
%O A013295 0,2
%A A013295 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A090407
%S A090407 1,6,136,2016,32896,523776,8390656,134209536,2147516416,34359607296,
%T A090407 549756338176,8796090925056,140737496743936,2251799780130816,
%U A090407 36028797153181696,576460751766552576,9223372039002259456
%N A090407 Sum{k=0..n, C(4n+1,4k) }.
%Y A090407 Cf. A070775, A001025, A090408, A038503.
%Y A090407 Sequence in context: A096756 A013299 A013295 this_sequence A075185 A003994 A053467
%Y A090407 Adjacent sequences: A090404 A090405 A090406 this_sequence A090408 A090409 A090410
%K A090407 easy,nonn
%O A090407 0,2
%A A090407 Paul Barry (pbarry(AT)wit.ie), Nov 29 2003
%I A075185
%S A075185 6,137,2436,40614,673470,11389140,198793980,3602823840,67991283360,
%T A075185 1337641905600,27440275262400,586731694348800,13067437397414400,
%U A075185 302870068070169600,7298072456298624000
%N A075185 One-fourth of fifth column of triangle A075181.
%C A075185 Also one-fourth of fifth diagonal of triangle A048594.
%F A075185 a(n)= A075181(n+5, 4)/4 = A048594(n+5, n+1)/4, n>=0.
%F A075185 a(n)= (n+1)!*S1(n+5, n+1)/4 with S1(n, m) := A008275(n, m) (Stirling1).
%Y A075185 Sequence in context: A013299 A013295 A090407 this_sequence A003994 A053467 A090944
%Y A075185 Adjacent sequences: A075182 A075183 A075184 this_sequence A075186 A075187 A075188
%K A075185 nonn,easy
%O A075185 0,1
%A A075185 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 19, 2002
%I A003994
%S A003994 6,138,1452,11444,78642,502846,3089624,18559208,110049502,647720562,
%T A003994 3796113284,22194147996,129581349642
%N A003994 Sequence b_4 (n) arising from homology of partitions with even number of blocks.
%D A003994 S. Sundaram, The homology of partitions with an even number of blocks, J. Alg. Comb., 4 (1995), 69-92.
%D A003994 S. Sundaram, Plethysm, partitions with an even number of blocks and Euler numbers, DIMACS Series, Vol. 24 (1996), 171-198, Amer. Math. Soc.
%p A003994 f := proc(n) option remember; if n = 1 then 2 else 3*f(n-1)+4*n-2; fi; end;
%Y A003994 Cf. A003993.
%Y A003994 Sequence in context: A013295 A090407 A075185 this_sequence A053467 A090944 A007340
%Y A003994 Adjacent sequences: A003991 A003992 A003993 this_sequence A003995 A003996 A003997
%K A003994 nonn
%O A003994 3,1
%A A003994 Sheila Sundaram (sheila(AT)paris-gw.cs.miami.edu)
%I A053467
%S A053467 1,6,138,22815,29197989,286181094816,21712697070199704,
%T A053467 12980080058620326927885,62082385554465497895132149640,
%U A053467 2405193620328895144597707267893468286
%N A053467 Number of directed 2-multigraphs on n nodes.
%Y A053467 Cf. A000273.
%Y A053467 Sequence in context: A090407 A075185 A003994 this_sequence A090944 A007340 A122483
%Y A053467 Adjacent sequences: A053464 A053465 A053466 this_sequence A053468 A053469 A053470
%K A053467 easy,nonn
%O A053467 1,2
%A A053467 Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 13 2000
%I A090944
%S A090944 1,6,140,270,672,1638,2970,6200,8190,18600,18620,27846,30240,32760,
%T A090944 55860,105664,117800,167400,173600,237510,242060,332640,360360,539400,
%U A090944 695520,726180,753480,1089270,1421280
%N A090944 Harmonic numbers (A001599) which are also arithmetic numbers (A003601).
%D A090944 T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.
%D A090944 R. K. Guy, Unsolved Problems in Number Theory, B2.
%Y A090944 Cf. A001599, A003601. Different from A090945.
%Y A090944 Sequence in context: A075185 A003994 A053467 this_sequence A007340 A122483 A123729
%Y A090944 Adjacent sequences: A090941 A090942 A090943 this_sequence A090945 A090946 A090947
%K A090944 nonn
%O A090944 1,2
%A A090944 njas, Feb 28 2004
%E A090944 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 01 2004
%I A007340 M4299
%S A007340 1,6,140,270,672,1638,2970,6200,8190,18600,18620,27846,30240,32760,55860,105664,
%T A007340 117800,167400,173600,237510,242060,332640,360360,539400,695520,726180,753480,
%U A007340 1089270,1421280,1539720,2229500,2290260,2457000
%N A007340 Integer average divisor divides the number. Or, both harmonic and arithmetic means of divisors are integral.
%C A007340 The following are also in A046985: 1,6,672,30240,32760. Also contains multiply perfect (A007691) numbers.
%D A007340 G. L. Cohen, personal communication.
%D A007340 O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.
%D A007340 N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with S. Plouffe), Academic Press, 1995.
%D A007340 D. Wells, Curious and interesting numbers, Penguin Books, p. 124.
%H A007340 Hisanori Mishima, Factorizations of many number sequences
%F A007340 a=Sigma[ 1, x ]/Sigma[ 0, x ] integer and b=x/a also.
%e A007340 x=270: Sigma[ 0,270 ]=16, Sigma[ 1,270 ]=720; average divisor a=720/16=45 and integer 45 divides x, x/a=270/45=6, but 270 is not in A007691.
%t A007340 Do[ a = DivisorSigma[0, n]/ DivisorSigma[1, n]; If[IntegerQ[n*a] && IntegerQ[1/a], Print[n]], {n, 1, 2500000}]
%Y A007340 Intersection of A003601 and A001599. Cf. A007691, A046985 - A046987, A046999.
%Y A007340 Sequence in context: A003994 A053467 A090944 this_sequence A122483 A123729 A123728
%Y A007340 Adjacent sequences: A007337 A007338 A007339 this_sequence A007341 A007342 A007343
%K A007340 nonn,nice
%O A007340 1,2
%A A007340 njas
%E A007340 Additional comments from Labos E. (labos(AT)ana.sote.hu)
%E A007340 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 03 2002
%I A122483
%S A122483 6,140,312,1560,14384,18018,40992,2337400,7012200,11027016,231402600
%N A122483 Numbers such that (-1)Sigma(m)*Sigma(m)= k*UnitaryPhi(m)*m for some integer k.
%C A122483 If both 2^n-3 and 2^n-1 are prime them numbers of the form 2^(n-1)*(M_n-2)*M_n appear on the sequence, where M_n means Mersenne prime.
%e A122483 2^8*7*19*37*73*509, 2^8*5*7*19*37*509, 2^8*5^2*7*19*29*31*37*509, 2^9*3*11*31*1021, 2^9*3*7*11^2*19*31*131*1021, 2^11*3^6*5*7*13*23*137*467*1093*4093, 2^13*3*11*43*127*16381, 2^13*3*7*11^2*19*43*127*131*16381 But between 3*10^7 and them, many terms may lack.
%Y A122483 Cf. A123124.
%Y A122483 Sequence in context: A053467 A090944 A007340 this_sequence A123729 A123728 A012785
%Y A122483 Adjacent sequences: A122480 A122481 A122482 this_sequence A122484 A122485 A122486
%K A122483 nonn
%O A122483 1,1
%A A122483 Yasutoshi Kohmoto zbi74583(AT)boat.zero.ad.jp, Sep 30 2006
%E A122483 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 01 2006
%I A123729
%S A123729 6,140,924,1560,14383,18018,22770,40992,93972,139672,142310,92628
%N A123729 Values of n associated with A123728.
%Y A123729 Cf. A123728.
%Y A123729 Sequence in context: A090944 A007340 A122483 this_sequence A123728 A012785 A012818
%Y A123729 Adjacent sequences: A123726 A123727 A123728 this_sequence A123730 A123731 A123732
%K A123729 nonn
%O A123729 1,1
%A A123729 Yasutoshi Kohmoto zbi74583(AT)boat.zero.ad.jp, Nov 18 2006
%I A123728
%S A123728 6,140,1316,1560,14384,18018,23562,40992,94188,141128,168730,187652
%N A123728 Call (m,n) a "(-1)SSU amicable pair" if (-1)Sigma(m)*Sigma(m) = k*UnitaryPhi(m)*(m+n) and (-1)Sigma(n)*Sigma(n) = k*UnitaryPhi(n)*(m+n) for some integer k. Sequence gives values of m, assumin n <= m.
%C A123728 a(3) is an example with m != n.
%e A123728 Kohmoto found the following terms.
%e A123728 k=3:
%e A123728 m=2^9*3*31*1021*7*23 n=2^9*3*31*1021*191
%e A123728 m=2^5*3*61*5*11 n=2^5*3*61*71
%e A123728 m=2^8*7*37*73*509*3*5 n=2^8*7*37*73*509*23
%e A123728 m=2^8*7*19*37*73*509*3*11 n=2^8*7*19*37*73*509*47
%e A123728 m=2^8*5*7*37*73*509*11*59 n=2^8*5*7*37*73*509*719
%e A123728 k=2:
%e A123728 m=2*3^2*5*13*23*29 n=2*3^2*5*13*719
%e A123728 m=2^2*7*3*11 n=2^2*7*47
%e A123728 m=3^2*5^2*13*29*17*19 n=3^2*5^2*13*29*359
%e A123728 k=5:
%e A123728 m=2^5*3^2*7*13*61*23*29 n=2^5*3^2*7*13*61*719
%e A123728 m=2^9*3^2*11*13*31*1021*23*29 n=2^9*3^2*11*13*31*1021*719
%Y A123728 Cf. A123729 (values of n), A123582 (values of k).
%Y A123728 Sequence in context: A007340 A122483 A123729 this_sequence A012785 A012818 A078450
%Y A123728 Adjacent sequences: A123725 A123726 A123727 this_sequence A123729 A123730 A123731
%K A123728 nonn
%O A123728 1,1
%A A123728 Yasutoshi Kohmoto zbi74583(AT)boat.zero.ad.jp, Nov 18 2006
%E A123728 R. J. Mathar did an exhaustive search up to 2000.
%E A123728 Giovanni Resta searched up to 10^7.
%I A012785
%S A012785 1,6,140,7616,731856,108552224,22943169600,6543956234752,
%T A012785 2420812240335104,1126850609820597760,644442125440553221120,
%U A012785 444134934522130204704768,363006744066808568769433600
%N A012785 tan(sec(x)*arcsin(x))=x+6/3!*x^3+140/5!*x^5+7616/7!*x^7+731856/9!*x^9...
%Y A012785 Sequence in context: A122483 A123729 A123728 this_sequence A012818 A078450 A059488
%Y A012785 Adjacent sequences: A012782 A012783 A012784 this_sequence A012786 A012787 A012788
%K A012785 nonn
%O A012785 0,2
%A A012785 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A012818
%S A012818 1,6,140,8456,957840,174293856,46514037440,17115342333056,
%T A012818 8304761365213440,5137820023434733056,3947233200665413667840,
%U A012818 3686931444647916864505856,4114664615209642258282352640
%N A012818 arctanh(sec(x)*sinh(x))=x+6/3!*x^3+140/5!*x^5+8456/7!*x^7...
%Y A012818 Sequence in context: A123729 A123728 A012785 this_sequence A078450 A059488 A067196
%Y A012818 Adjacent sequences: A012815 A012816 A012817 this_sequence A012819 A012820 A012821
%K A012818 nonn
%O A012818 0,2
%A A012818 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A078450
%S A078450 1,6,140,10296,1560090,818269800,890504276970,578105086816530,
%T A078450 845098382127164340,1345577210752717337010,2349420395175736366400280,
%U A078450 4072812804055080385050520770,16954222832305267550769863845710
%N A078450 a(n) = product of terms in n-th row of A078448.
%o A078450 (PARI) {s=[1]; for (n=2,14,print1(prod(i=1,n-1,s[i]),","); a=s[n-1]; s=[a+1]; for(j=2,n,k=s[j-1]+1; c=1; while(c>0,b=1; for(i=1,matsize(s)[2],if(gcd(k,s[i])>1,b=0)); if(b==0,k++,c=0)); s=concat(s,k)))}
%Y A078450 Cf. A078447, A078448, A078449.
%Y A078450 Sequence in context: A123728 A012785 A012818 this_sequence A059488 A067196 A048863
%Y A078450 Adjacent sequences: A078447 A078448 A078449 this_sequence A078451 A078452 A078453
%K A078450 nonn
%O A078450 1,2
%A A078450 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 27 2002
%E A078450 Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Dec 10 2002
%I A059488
%S A059488 1,6,140,12936,4756752,6974908512,40825196325056,954410297723625600,
%T A059488 89149543741372647686400,33280303224443643993143232000,49660896290963321355372907102080000,
%U A059488 296248087478941460167300263349693113600000,7065635944743752671502919147104799866118720000000
%N A059488 Expansion of generating function A_{UU}^(2)(4n;2,1,1).
%H A059488 G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv math.CO/0008184 [Th. 5]
%p A059488 A059488 := proc(n) local i, j, t1; t1 := 2^(n^2 + 2*n); for i to 2*n + 1 do for j to 2*n + 1 do if i mod 2 <> 0 and j mod 2 = 0 then t1 := t1*(2*j - 2*i + 1)/(2*j - 2*i) end if end do end do; t1 end proc;
%Y A059488 Sequence in context: A012785 A012818 A078450 this_sequence A067196 A048863 A111839
%Y A059488 Adjacent sequences: A059485 A059486 A059487 this_sequence A059489 A059490 A059491
%K A059488 nonn,easy
%O A059488 0,2
%A A059488 njas, Feb 04 2001
%I A067196
%S A067196 1,6,142,154,157,167,168,169,209,213,214,231,232,235,236,238,239,240,
%T A067196 242,243,244,245,247,248,251,252,257,259,260,261,263,264,266,269,270,
%U A067196 278,279,280,301,318,362,363,364,366,367,368,369,371,372,391,392,402
%N A067196 Numbers n such that M(n)=sum(i=1,n,mu(phi(i))) where M(n) is the Mertens function A002321(n).
%Y A067196 Sequence in context: A012818 A078450 A059488 this_sequence A048863 A111839 A128785
%Y A067196 Adjacent sequences: A067193 A067194 A067195 this_sequence A067197 A067198 A067199
%K A067196 nonn
%O A067196 1,2
%A A067196 Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 19 2002
%I A048863
%S A048863 1,1,1,6,142,2518,49836,1012859,24211838,721500294,22627459401,844130935668,
%T A048863 34729870646918
%N A048863 Number of nonprimes (1 and composites) in the reduced residue system of n-th primorial number (A002110).
%F A048863 A005867(n) - [ A000849(n) - A001221(A002110(n)) ] = A000010(A002110(n)) - [ A000720((A002110(n)) - n ]
%e A048863 n=3, 3rd primorial is 30, EulerPhi(30)=8, a(3)=1 since 1 is regarded here a nonprime. See A048597. n=4, 4th primorial is 210, size of its reduced residue system(RRS) is 48 of 6 is either composite or 1: {1,121,143,169,187,209}.
%Y A048863 Cf. A002110, A000010, A005867, A000720, A048597, A007625, A048862.
%Y A048863 A048683(n) = A005867(n) -A000849(n) + n.
%Y A048863 Sequence in context: A078450 A059488 A067196 this_sequence A111839 A128785 A010043
%Y A048863 Adjacent sequences: A048860 A048861 A048862 this_sequence A048864 A048865 A048866
%K A048863 more,nonn
%O A048863 1,4
%A A048863 Labos E. (labos(AT)ana.sote.hu)
%I A111839
%S A111839 0,1,6,142,31800,159468264,2481298801008,1414130111428687344,
%T A111839 1827317023092830201950080,89946874545119714361987192509568,
%U A111839 9262235489215916508714844705185660161280
%V A111839 0,1,-6,142,31800,-159468264,-2481298801008,1414130111428687344,
%W A111839 1827317023092830201950080,-89946874545119714361987192509568,
%X A111839 -9262235489215916508714844705185660161280
%N A111839 Column 0 of the matrix logarithm (A111838) of triangle A111835, which shifts columns left and up under matrix 8-th power; these terms are the result of multiplying the element in row n by n!.
%C A111839 Let q=8; the g.f. of column k of A111825^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).
%F A111839 E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(8^j*x)/(j+1).
%e A111839 A(x) = x - 6/2!*x^2 + 142/3!*x^3 + 31800/4!*x^4 - 159468264/5!*x^5 +...
%e A111839 where e.g.f. A(x) satisfies:
%e A111839 x/(1-x) = A(x) + A(x)*A(8*x)/2! + A(x)*A(8*x)*A(8^2*x)/3! +
%e A111839 A(x)*A(8*x)*A(8^2*x)*A(8^3*x)/4! + ...
%e A111839 Let G(x) be the g.f. of A111836 (column 1 of A111835), then
%e A111839 G(x) = 1 + 8*A(x) + 8^2*A(x)*A(8*x)/2! +
%e A111839 8^3*A(x)*A(8*x)*A(8^2*x)/3! +
%e A111839 8^4*A(x)*A(8*x)*A(8^2*x)*A(8^3*x)/4! + ...
%o A111839 (PARI) {a(n,q=8)=local(A=x/(1-x+x*O(x^n)));for(i=1,n, A=x/(1-x)/(1+sum(j=1,n,prod(k=1,j,subst(A,x,q^k*x))/(j+1)!))); return(n!*polcoeff(A,n))}
%Y A111839 Cf. A111835 (triangle), A111836, A111838 (matrix log); A110505 (q=-1), A111814 (q=2), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7).
%Y A111839 Sequence in context: A059488 A067196 A048863 this_sequence A128785 A010043 A085905
%Y A111839 Adjacent sequences: A111836 A111837 A111838 this_sequence A111840 A111841 A111842
%K A111839 sign
%O A111839 0,3
%A A111839 Gottfried Helms (helms(AT)uni-kassel.de) and Paul D. Hanna (pauldhanna(AT)juno.com), Aug 22 2005
%I A128785
%S A128785 0,6,144,1944,20736,194400,1679616,13716864,107495424,816293376,
%T A128785 6046617600,43898443776,313456656384,2207257288704,15359376162816,
%U A128785 105791621529600,722204136308736,4891804579528704,32905425960566784
%N A128785 n^2*6^n.
%Y A128785 Cf. A036289; A007758.
%Y A128785 Sequence in context: A067196 A048863 A111839 this_sequence A010043 A085905 A090443
%Y A128785 Adjacent sequences: A128782 A128783 A128784 this_sequence A128786 A128787 A128788
%K A128785 nonn
%O A128785 0,2
%A A128785 Mohammad K. Azarian (azarian(AT)evansville.edu), Apr 07 2007
%I A010043
%S A010043 0,6,144,3480,95616,2995296,106308864,4224923520,186254217216,
%T A010043 9025003101696,477123118608384,27334467671746560,1687499455653052416,
%U A010043 111689161007888080896,7890849624188124463104
%N A010043 High-temperature expansion of susceptibility mu_2 for cubic lattice.
%D A010043 M. Luescher and P. Weisz, Application of the linked cluster expansion to the n-component phi^4 theory, Nuclear Physics B 300 (1988), 325-359.
%Y A010043 Sequence in context: A048863 A111839 A128785 this_sequence A085905 A090443 A133460
%Y A010043 Adjacent sequences: A010040 A010041 A010042 this_sequence A010044 A010045 A010046
%K A010043 nonn
%O A010043 0,2
%A A010043 njas
%I A085905
%S A085905 1,6,144,5952,772560,73664640,29745273600,8715934402560,
%T A085905 5068085799813120,2756328707949465600,4581860819083475558400,
%U A085905 2696083278990328597708800,7844679216026128507826995200
%N A085905 Permanent of the symmetric n X n matrix M defined by M(i,j) = lcm(i,j) for 1 <= i,j <= n.
%e A085905 a(2)=6 since the 2 by 2 matrix A with rows [1,2],[2,2] has permanent 1*2+2*2=6.
%p A085905 with(linalg): a:=(i,j)->lcm(i,j): seq(permanent(matrix(n,n,a)),n=1..14); (Deutsch)
%o A085905 (PARI) permRWNb(a)=n=matsize(a)[1];if(n==1,return(a[1,1]));sg=1;in=vectorv(n);x=in;x=a[,n]-sum(j=1,n,a[,j])/2;p=prod(i=1,n,x[i]);for(k=1,2^(n-1)-1,sg=-sg;j=valuation(k,2)+1;z=1-2*in[j];in[j]+=z;x+=z*a[,j];p+=prod(i=1,n,x[i],sg));return(2*(2*(n%2)-1)*p) for(n=1,20,a=matrix(n,n,i,j,lcm(i,j));print1(permRWNb(a)",")) - Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007
%Y A085905 Cf. A060238, A085244, A034444.
%Y A085905 Sequence in context: A111839 A128785 A010043 this_sequence A090443 A133460 A041271
%Y A085905 Adjacent sequences: A085902 A085903 A085904 this_sequence A085906 A085907 A085908
%K A085905 nonn
%O A085905 1,2
%A A085905 Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003
%E A085905 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2005
%I A090443
%S A090443 1,6,144,8640,1036800,217728000,73156608000,36870930432000,
%T A090443 26547069911040000,26281599211929600000,34691710959747072000000,
%U A090443 59530976006925975552000000,130015651599126330605568000000
%N A090443 Fourth column (m=3) of triangle A090441.
%F A090443 a(n)= (n+2)!*(n+1)!*n!/2, n>=0.
%p A090443 a:=n->mul(j^3-j, j=2..n): seq(a(n), n=1..13); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2008
%Y A090443 Cf. A010790, A090444.
%Y A090443 Sequence in context: A128785 A010043 A085905 this_sequence A133460 A041271 A121473
%Y A090443 Adjacent sequences: A090440 A090441 A090442 this_sequence A090444 A090445 A090446
%K A090443 nonn,easy
%O A090443 0,2
%A A090443 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 23 2003
%I A133460
%S A133460 1,6,144,13824,5308416,8153726976,50096498540544,1231171548132409344,
%T A133460 121029087867608368152576,47590573814949492091483324416,
%U A133460 74853500292876717928978827574247424
%N A133460 3^n*2^(n^2).
%C A133460 Hankel transform of A089022.
%F A133460 a(n)=3^n*2^(n^2)=A000244(n)*A002416(n).
%Y A133460 Sequence in context: A010043 A085905 A090443 this_sequence A041271 A121473 A063419
%Y A133460 Adjacent sequences: A133457 A133458 A133459 this_sequence A133461 A133462 A133463
%K A133460 easy,nonn
%O A133460 0,2
%A A133460 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 28 2007
%I A041271
%S A041271 1,6,145,876,21169,127890,3090529,18671064,451196065,2725847454,
%T A041271 65871534961,397955057220,9616792908241,58098712506666,
%U A041271 1403985893068225,8482014070916016,204972323595052609,1238315955641231670
%N A041271 Denominators of continued fraction convergents to sqrt(148).
%Y A041271 Cf. A041270.
%Y A041271 Sequence in context: A085905 A090443 A133460 this_sequence A121473 A063419 A099185
%Y A041271 Adjacent sequences: A041268 A041269 A041270 this_sequence A041272 A041273 A041274
%K A041271 nonn,cofr,easy
%O A041271 0,2
%A A041271 njas
%I A121473
%S A121473 0,1,6,146,8,37783544111994270385152,
%T A121473 784637716923335095479473680436259502469253233551410733056,
%U A121473 309485009821345068724781056
%N A121473 Partial quotients of the continued fraction expansion of the constant A121472 defined by the sums: c = Sum_{n>=1} 1/2^[log_2(e^n)] = Sum_{n>=1} [log(2^n)]/2^n.
%C A121473 A "devil's staircase" type of constant has large partial quotients in its continued fraction expansion. See MathWorld link for more information.
%H A121473 Eric Weisstein's World of Mathematics, Devil's Staircase
%e A121473 c=0.857282383103406177511903308509733997590988312093146922257824...
%e A121473 The number of 1's in the binary expansion of a(n) is given by
%e A121473 the partial quotients of continued fraction of log(2):
%e A121473 log(2) = [0; 1, 2, 3, 1, 6, 3, 1, 1, 2, 1, 1, 1, 1, 3, 10, ...]
%e A121473 as can be seen by the binary expansions of a(n):
%e A121473 a(0) = 0
%e A121473 a(1) = 2^0
%e A121473 a(2) = 2^2 + 2^1
%e A121473 a(3) = 2^7 + 2^4 + 2^1
%e A121473 a(4) = 2^3
%e A121473 a(5) = 2^75 + 2^62 + 2^49 + 2^36 + 2^23 + 2^10
%e A121473 a(6) = 2^189 + 2^101 + 2^13
%e A121473 a(7) = 2^88
%e A121473 a(8) = 2^277
%e A121473 a(9) = 2^1007 + 2^365
%e A121473 a(10) = 2^642
%e A121473 a(11) = 2^1649
%e A121473 a(12) = 2^2291
%e A121473 a(13) = 2^3940
%e A121473 a(14) = 2^26573 + 2^16402 + 2^6231
%Y A121473 Cf. A121472 (constant), A121474 (dual constant), A121475.
%Y A121473 Sequence in context: A090443 A133460 A041271 this_sequence A063419 A099185 A065986
%Y A121473 Adjacent sequences: A121470 A121471 A121472 this_sequence A121474 A121475 A121476
%K A121473 cofr,nonn
%O A121473 0,3
%A A121473 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 01 2006
%I A063419
%S A063419 1,6,146,4332,135954,4395456,144840476,4836766584,163112472594,
%T A063419 5542414273884,189456975899496,6507792553644256,224442843729333276,
%U A063419 7766945604528200460,269557528994032024080,9378595792117360310832
%N A063419 Central sextinomial coefficients.
%C A063419 Largest coefficient of sum(x^j,j=0..5)^(2*n). a(n)= A018901(2*n).
%F A063419 a(n)= A063260(2*n, 5*n)= [x^(5*n)]sum(x^j, j=0..5)^(2*n).
%F A063419 a(n) = sum((-1)^(k)*binomial(2*n,k)*binomial(7*n-6*k-1, 2*n-1), k=0..floor(5*n/6)) - Warut Roonguthai (warut822(AT)yahoo.com), May 22 2006
%Y A063419 Central q-nomial coefficients (appearing once) for q=2..5: A000984, A002426, A005721, A005191. For q=7: A025012.
%Y A063419 Sequence in context: A133460 A041271 A121473 this_sequence A099185 A065986 A012791
%Y A063419 Adjacent sequences: A063416 A063417 A063418 this_sequence A063420 A063421 A063422
%K A063419 nonn,easy
%O A063419 0,2
%A A063419 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 24 2001
%I A099185
%S A099185 1,6,146,2074806,5954444157018557346,
%T A099185 140744820294208035204656447906095566299588102457814757606,
%U A099185 1858685896365056640452604182778243755878210128325493631436394942328487801924642707284183892998994140529418479176238949509991689659112331788418314832322689820822344586546
%N A099185 Iterated octahedral numbers, starting at oct(2) = 6.
%C A099185 This need not start at oct(2) = 6. For example, if a(1) = oct(3) = 19, then a(4)= oct(19) = 4579; a(5) = oct(4579) = 64005999219; a(6) = oct(64005999219) = 174811816875659072517015216413379.
%D A099185 Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 50, 1996.
%D A099185 Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.
%D A099185 J. V. Post, "Iterated Triangular numbers", preprint.
%H A099185 J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
%H A099185 Eric Weisstein's World of Mathematics, "Octahedral Number."
%F A099185 Given the octahedral number formula oct(n) = (2*n^3 + n)/3, define: a(0, n) = 0; a(1, n) = oct(n); a(2, n) = oct(oct(n)); in general for k>0 a(k+1, n) = oct(a(k, n)); the octahedral number of the octahedral number of ... of n. a(n) = 2*a(n-1) + 3; generating function = 1/(exp(x)-1).
%e A099185 a(3) = 2074806 because a(1) = the 2nd octahedral number = oct(2) = 6; a(2) = oct(oct(2)) = the 6th octahedral number = oct(6) = (2*6^3 + 6)/3 = 146; a(3) = oct(oct(oct(2))) = the 146th octahedral number = oct(146) = (2*146^3 + 146)/3 = 2074806.
%Y A099185 Cf. A007501, A005900.
%Y A099185 Sequence in context: A041271 A121473 A063419 this_sequence A065986 A012791 A089480
%Y A099185 Adjacent sequences: A099182 A099183 A099184 this_sequence A099186 A099187 A099188
%K A099185 easy,nonn,uned
%O A099185 0,2
%A A099185 Jonathan Vos Post (jvospost2(AT)yahoo.com), Nov 15 2004
%I A065986
%S A065986 6,147,286,376,534,738,805,2392,2406,4324,8214,9606,10362,12126,16263,
%T A065986 17511,27639,29151,39215,48616,60687,61132,61915
%N A065986 Numbers n such that Sigma(n) = EulerPhi(n+1) + EulerPhi(n) + EulerPhi(n-1).
%e A065986 Sigma(6) = 12 = 6 + 2 + 4 = EulerPhi(7) + EulerPhi(6) + EulerPhi(5).
%Y A065986 Cf. A000010.
%Y A065986 Sequence in context: A121473 A063419 A099185 this_sequence A012791 A089480 A056427
%Y A065986 Adjacent sequences: A065983 A065984 A065985 this_sequence A065987 A065988 A065989
%K A065986 nonn
%O A065986 1,1
%A A065986 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 10 2001
%I A012791
%S A012791 1,6,148,9184,1066640,199000736,54431471040,20523724218880,
%T A012791 10203766053312768,6467780044492713472,5090997086675585360896,
%U A012791 4871968356294907783585792,5570589015199967686042095616
%N A012791 arctanh(sec(x)*arcsin(x))=x+6/3!*x^3+148/5!*x^5+9184/7!*x^7...
%Y A012791 Sequence in context: A063419 A099185 A065986 this_sequence A089480 A056427 A056418
%Y A012791 Adjacent sequences: A012788 A012789 A012790 this_sequence A012792 A012793 A012794
%K A012791 nonn
%O A012791 0,2
%A A012791 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A089480
%S A089480 1,6,150,6,18,13032,1440,4992,672,1440,288,576,0,24,0,96,3513720,693840,
%T A089480 2626800,604200,1451400,468000,962400,252000,425400,190800,379200,97200,
%U A089480 205440,100800,132000,28800,108000,28800,44400,33600,61200,9600,14400,0
%N A089480 Triangle T(n,k) read by rows, where T(n,k) = number of times the permanent of a real nonsingular n X n (0,1)-matrix takes the value k, for n >= 1, 1 <= k <= A000255(n).
%C A089480 This sequence was first provided by Jaap Spies (j.spies(AT)hccnet.nl).
%Y A089480 T(n, A000255(n)) = A052655(n). The n-th row of the table contains A089475(n) nonzero entries. Cf. A089479 occurrence counts for permanents of all (0, 1)-matrices, A089481 occurrence counts for permanents of singular (0, 1)-matrices.
%Y A089480 Sequence in context: A099185 A065986 A012791 this_sequence A056427 A056418 A070025
%Y A089480 Adjacent sequences: A089477 A089478 A089479 this_sequence A089481 A089482 A089483
%K A089480 nonn,tabf
%O A089480 1,2
%A A089480 Hugo Pfoertner (hugo(AT)pfoertner.org), Nov 04 2003
%I A056427
%S A056427 0,0,0,0,6,150,400,4080,15480,127812,269340,3493530,5777190,57262050,
%T A056427 210945176,1030384035,2493913170,32176432430,51785999300,562228198086,
%U A056427 1805427491520,10438821574410,22865672706000
%N A056427 Number of primitive (period n) step cyclic shifted sequences using exactly five different symbols.
%C A056427 See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.
%D A056427 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%F A056427 sum mu(d)*A056418(n/d) where d|n.
%Y A056427 Cf. A056422.
%Y A056427 Sequence in context: A065986 A012791 A089480 this_sequence A056418 A070025 A065946
%Y A056427 Adjacent sequences: A056424 A056425 A056426 this_sequence A056428 A056429 A056430
%K A056427 nonn
%O A056427 1,5
%A A056427 Marks R. Nester (nesterm(AT)dpi.qld.gov.au)
%I A056418
%S A056418 0,0,0,0,6,150,400,4080,15480,127818,269340,3493680,5777190,57262450,
%T A056418 210945182,1030388115,2493913170,32176448060,51785999300,562228325904,
%U A056418 1805427491920,10438821843750,22865672706000
%N A056418 Number of step cyclic shifted sequences using exactly five different symbols.
%C A056418 See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.
%D A056418 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%F A056418 A056413(n)-5*A056412(n)+10*A056411(n)-10*A002729(n)+5.
%Y A056418 Cf. A056413.
%Y A056418 Sequence in context: A012791 A089480 A056427 this_sequence A070025 A065946 A013296
%Y A056418 Adjacent sequences: A056415 A056416 A056417 this_sequence A056419 A056420 A056421
%K A056418 nonn
%O A056418 1,5
%A A056418 Marks R. Nester (nesterm(AT)dpi.qld.gov.au)
%I A070025
%S A070025 6,150,2730,9000,9240,35280,41760,43050,53280,65520,76650,96180,111030,
%T A070025 148200,197370,207480,213360,226380,254280,264600,309480,332160,342450,
%U A070025 352740,375450,381990,440550,458790,501030,527070,552030,642360,660810
%N A070025 At these values of n the first, 2nd, 3rd and 4th cyclotomic polynomials all give prime numbers.
%F A070025 n-1, n+1, 1+n+n^2 and 1+n^2 are all primes.
%e A070025 n=6: 5,7,43 and 37 are prime values of first 4 cyclotomic polynomials.
%t A070025 lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[1+n+n^2]&&PrimeQ[1+n^2],AppendTo[lst,n]],{n,10^6}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 19 2008]
%Y A070025 Cf. A070155-A070157, A000068, A006313-A006316, A056993-A056995, A005574, A057465, A057002, A070020, A070042.
%Y A070025 Sequence in context: A089480 A056427 A056418 this_sequence A065946 A013296 A013301
%Y A070025 Adjacent sequences: A070022 A070023 A070024 this_sequence A070026 A070027 A070028
%K A070025 easy,nonn
%O A070025 1,1
%A A070025 Labos E. (labos(AT)ana.sote.hu), May 07 2002
%I A065946
%S A065946 0,0,6,150,3870,110670,3538500,125941284,4953759300,213744815460,
%T A065946 10047637214010,511403305348650,28029852267603186,1646397200571955650,
%U A065946 103190849406195456360,6875135229835376875560,485256294032090950981800
%V A065946 0,0,6,-150,3870,-110670,3538500,-125941284,4953759300,-213744815460,
%W A065946 10047637214010,-511403305348650,28029852267603186,-1646397200571955650,
%X A065946 103190849406195456360,-6875135229835376875560,485256294032090950981800
%N A065946 Bessel polynomial {y_n}''(-2).
%D A065946 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A065946 Index entries for sequences related to Bessel functions or polynomials
%Y A065946 Cf. A001518, A001516.
%Y A065946 Sequence in context: A056427 A056418 A070025 this_sequence A013296 A013301 A089482
%Y A065946 Adjacent sequences: A065943 A065944 A065945 this_sequence A065947 A065948 A065949
%K A065946 sign
%O A065946 0,3
%A A065946 njas, Dec 08 2001
%I A013296
%S A013296 1,6,150,7560,642600,82328400,14799985200,3550699152000,
%T A013296 1095489931536000,422416533258720000,199001229578030880000,
%U A013296 112442434205652270720000,75042084306026165328000000
%N A013296 tan(log(x+1)-arctanh(x))=-1/2!*x^2-6/4!*x^4-150/6!*x^6-7560/8!*x^8...
%Y A013296 Sequence in context: A056418 A070025 A065946 this_sequence A013301 A089482 A126679
%Y A013296 Adjacent sequences: A013293 A013294 A013295 this_sequence A013297 A013298 A013299
%K A013296 nonn
%O A013296 0,2
%A A013296 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A013301
%S A013301 1,6,150,7560,650160,84823200,15617281680,3855823171200,
%T A013301 1229340262550400,491641405006752000,241000527467642342400,
%U A013301 142107676844443620710400,99236357585615999548800000
%N A013301 arctanh(log(x+1)-atanh(x))=-1/2!*x^2-6/4!*x^4-150/6!*x^6-7560/8!*x^8...
%Y A013301 Sequence in context: A070025 A065946 A013296 this_sequence A089482 A126679 A003766
%Y A013301 Adjacent sequences: A013298 A013299 A013300 this_sequence A013302 A013303 A013304
%K A013301 nonn
%O A013301 0,2
%A A013301 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A089482
%S A089482 1,6,150,13032,3513720,2722682160
%N A089482 Number of real {0,1}-matrices having permanent=1.
%C A089482 a(6) from Gordon Royle (gordon(AT)csse.uwa.edu.au).
%e A089482 a(2)=6 because there are 6 matrices ((1,0),(0,1)), ((0,1),(1,0)), ((0,1),(1,1)), ((1,0),(1,1,)), ((1,1),(0,1)), ((1,1,),(1,0)) with permanent=1.
%Y A089482 Cf. A088672 number of (0, 1)-matrices with zero permanent, A089479 occurrence counts for permanents of all (0, 1)-matrices, A089480 occurrence counts for permanents of non-singular (0, 1)-matrices.
%Y A089482 Sequence in context: A065946 A013296 A013301 this_sequence A126679 A003766 A046182
%Y A089482 Adjacent sequences: A089479 A089480 A089481 this_sequence A089483 A089484 A089485
%K A089482 more,nonn
%O A089482 1,2
%A A089482 Hugo Pfoertner (hugo(AT)pfoertner.org), Nov 05 2003
%I A126679
%S A126679 1,6,150,13500,4063500,3925341000,11874156525000,110785880378250000,3157508376660503250000,
%T A126679 273206569798926704209500000,71477668823644198988810437500000,56393736371790563676201770874375000000,
%U A126679 133940819650376139577910502205498936875000000,956563276525616170757609342853980880495071250000000
%N A126679 Product_{i=3..n} Stirling_2(i,3).
%Y A126679 Partial products of A000392.
%Y A126679 Sequence in context: A013296 A013301 A089482 this_sequence A003766 A046182 A092122
%Y A126679 Adjacent sequences: A126676 A126677 A126678 this_sequence A126680 A126681 A126682
%K A126679 nonn
%O A126679 3,2
%A A126679 njas, Feb 13 2007
%I A003766
%S A003766 6,152,1608,15420,127980,1003360,7432708,53294540,371397240,
%T A003766 2537155684,17047659916,113102692016,742597784164,4835184613212,
%U A003766 31267479066856,201066698078244,1286998671857356,8206523391863296
%N A003766 Number of Hamiltonian paths in W_4 X P_n.
%H A003766 F. Faase, Counting Hamilton cycles in product graphs
%F A003766 Faase gives a 16-term linear recurrence on his web page.
%Y A003766 Sequence in context: A013301 A089482 A126679 this_sequence A046182 A092122 A003460
%Y A003766 Adjacent sequences: A003763 A003764 A003765 this_sequence A003767 A003768 A003769
%K A003766 nonn
%O A003766 1,1
%A A003766 Frans Faase (Frans_LiXia(AT)wxs.nl)
%I A046182
%S A046182 1,6,153,638,15041,62566,1473913,6130878,144428481,600763526,
%T A046182 14152517273,58868694718,1386802264321,5768531318886,135892469386233,
%U A046182 565257200556158,13316075197586561,55389437123184646
%N A046182 Indices of triangular numbers which are also octagonal.
%H A046182 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%F A046182 For n odd, a(n+2)=98*a(n+1)-a(n)+48; for n even, a(n+1)=49*a(n)+24+10*(24*a(n)^2+24*a(n)+16)^0.5 - Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 03 2007, Oct 09 2007
%Y A046182 Cf. A046181, A046183, A046190.
%Y A046182 Cf. A046181 A046183.
%Y A046182 Sequence in context: A089482 A126679 A003766 this_sequence A092122 A003460 A128120
%Y A046182 Adjacent sequences: A046179 A046180 A046181 this_sequence A046183 A046184 A046185
%K A046182 nonn
%O A046182 1,2
%A A046182 Eric Weisstein (eric(AT)weisstein.com)
%E A046182 More terms from Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 03 2007
%I A092122
%S A092122 6,154,310,370,2829,3526,15320,20462
%N A092122 Let R_{k}(n) = the digit reversal of n in base k (R_{k}(n) is written in base 10). Sequence gives numbers n such that n = Sum_{d|n, d>1} R_{d}(n).
%e A092122 If n=154: Sum_{d|154, d>1} R_{d}(154) = 89 + 10 + 34 + 11 + 7 + 2 + 1 = 154.
%Y A092122 Cf. A004086, A030101-A030108, A056960-A056963.
%Y A092122 Sequence in context: A126679 A003766 A046182 this_sequence A003460 A128120 A030449
%Y A092122 Adjacent sequences: A092119 A092120 A092121 this_sequence A092123 A092124 A092125
%K A092122 more,nonn,base
%O A092122 1,1
%A A092122 Naohiro Nomoto (pcmusume(AT)m11.alpha-net.ne.jp), Mar 30 2004
%I A003460 M4300
%S A003460 1,6,154,66344,15471166144,663447306235471066144,
%T A003460 1547116614473162154311663447306215471066144
%N A003460 Octal formula for dragon curve of order n.
%D A003460 M. Gardner, Mathematical Games, Sci. Amer. Vol. 216 (No. 4, Apr. 1967), p. 118.
%D A003460 M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 216.
%H A003460 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%Y A003460 Sequence in context: A003766 A046182 A092122 this_sequence A128120 A030449 A120277
%Y A003460 Adjacent sequences: A003457 A003458 A003459 this_sequence A003461 A003462 A003463
%K A003460 nonn
%O A003460 1,2
%A A003460 njas
%I A128120
%S A128120 6,156,430,1602,5365,7668,7669,16249,16303,51520,65439,73648,90663,
%T A128120 114080,139141,152713,154708,154709,158457,160588,165441,188040,188716,
%U A128120 189278,250459,251435,270415,271426,272816,298223,316787,331003,347609
%N A128120 Numbers n such that n-th and (n+1)-th primes are in A125146.
%Y A128120 Cf. A125146.
%Y A128120 Sequence in context: A046182 A092122 A003460 this_sequence A030449 A120277 A015086
%Y A128120 Adjacent sequences: A128117 A128118 A128119 this_sequence A128121 A128122 A128123
%K A128120 nonn
%O A128120 1,1
%A A128120 Zak Seidov (zakseidov(AT)yahoo.com), May 02 2007
%I A030449
%S A030449 1,6,159,332380,2751884514765,272622932796281408879065986,
%T A030449 3641839910835401567626683593436003894250931310990279691
%N A030449 Number of elements in the free band (idempotent semigroup) on n generators.
%C A030449 An idempotent semigroup satisfies the equation xx=x for any element x.
%D A030449 J. Howie, Fundamentals of Semigroup Theory, Oxford University Press 1995, p. 123.
%H A030449 Index entries for sequences related to semigroups
%F A030449 a_n=\Sum_{k=1}^{n} C(n, k) A030450(k).
%Y A030449 Cf. A030450. A005345(n)=a(n)+1.
%Y A030449 Sequence in context: A092122 A003460 A128120 this_sequence A120277 A015086 A052466
%Y A030449 Adjacent sequences: A030446 A030447 A030448 this_sequence A030450 A030451 A030452
%K A030449 nonn
%O A030449 1,2
%A A030449 marcel_j(AT)hilbert.maths.utas.edu.au (Marcel Jackson)
%I A120277
%S A120277 6,160,2842,44868,681604,10248992,154149762,2327405740,35305388536,
%T A120277 538000530912,8231764528156,126399786937760,1946868985459272,
%U A120277 30066806831424448,465425347391123282,7219408741591089660
%N A120277 Sum of all matrix elements of n X N matrix M[i,j]=(2n+i+j)!/(n+i)!/(n+j)!, i,j=1..n.
%C A120277 p divides a((p-1)/2) for prime p>2.
%F A120277 a(n) = Sum[Sum[(2n+i+j)!/(n+i)!/(n+j)!,{i,1,n}],{j,1,n}].
%t A120277 Table[Sum[Sum[(2n+i+j)!/(n+i)!/(n+j)!,{i,1,n}],{j,1,n}],{n,1,20}]
%Y A120277 Sequence in context: A003460 A128120 A030449 this_sequence A015086 A052466 A078535
%Y A120277 Adjacent sequences: A120274 A120275 A120276 this_sequence A120278 A120279 A120280
%K A120277 nonn
%O A120277 1,1
%A A120277 Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 04 2006
%I A015086
%S A015086 1,1,6,161,20466,12833546,40130703276,627122621447281,
%T A015086 48995209411107768186,19138851672289046707772366,
%U A015086 37380607950584029444762130426196
%N A015086 q-Catalan numbers (recurrence version) for q=5.
%F A015086 a(n) = sum_{i=1}^{n-1} q^{(i-1)} a(i) a(n-i).
%Y A015086 Sequence in context: A128120 A030449 A120277 this_sequence A052466 A078535 A104729
%Y A015086 Adjacent sequences: A015083 A015084 A015085 this_sequence A015087 A015088 A015089
%K A015086 nonn
%O A015086 1,3
%A A015086 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A052466
%S A052466 6,162,1007,27371,170176,4625692,28759737,781741941,4860395546,
%T A052466 132114388022,821406847267,22327331575711,138817757188116,
%U A052466 3773319036295152,23460200964791597,637690917133880681
%N A052466 a(n) is the solution k to Mod[24k,13^n]==1.
%C A052466 Related to a generalization of a Ramanujan congruence for the partition function P.
%H A052466 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%t A052466 Table[PowerMod[24, -1, 13^d], {d, 20}]
%Y A052466 Cf. A052462, A052464, A052465.
%Y A052466 Sequence in context: A030449 A120277 A015086 this_sequence A078535 A104729 A106661
%Y A052466 Adjacent sequences: A052463 A052464 A052465 this_sequence A052467 A052468 A052469
%K A052466 nonn,easy
%O A052466 1,1
%A A052466 Eric Weisstein (eric(AT)weisstein.com)
%I A078535
%S A078535 1,6,162,5760,232254,10077696,458960580,21634449408,1046465787510,
%T A078535 51644846702592,2590092194793948,131621703842267136,
%U A078535 6762649550214036780
%N A078535 Coefficients of power series that satisfies A(x)^6 - 36x*A(x)^7 = 1, A(0)=1.
%C A078535 If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2) (conjecture).
%C A078535 If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k)=n^(2k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - Emeric Deutsch, Dec 10 2002
%C A078535 A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n -(n^2)*x*A(x)^(n+1) = 1 reduces to A(x)=1+xA(x)^2. - Emeric Deutsch, Dec 10 2002
%F A078535 a(n)=6^(2n)*binomial(7n/6-5/6, n)/(n+1) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2002
%e A078535 A(x)^6 - 36x*A(x)^7 = 1 since A(x)^6 = 1 +36x +1512x^2 +68040x^3 +3193344x^4 +... and A(x)^7 = 1 +42x +1890x^2 +88704x^3 +... also a(5)=6^9, a(11)=6^22 = 131621703842267136.
%Y A078535 Cf. A078531, A078532, A078533, A078534.
%Y A078535 Sequence in context: A120277 A015086 A052466 this_sequence A104729 A106661 A003720
%Y A078535 Adjacent sequences: A078532 A078533 A078534 this_sequence A078536 A078537 A078538
%K A078535 nonn
%O A078535 0,2
%A A078535 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 28 2002
%I A104729
%S A104729 6,166,4666,46666,616666,5666666,62666666,166666666,6466666666,
%T A104729 46666666666,666626666666,6666566666666,66266666666666,646666666666666,
%U A104729 2666666666666666,46666666666666666,666566666666666666
%N A104729 Smallest semiprime containing exactly n 6's.
%e A104729 a(2)=166 because 166 is the smallest semiprime containing exactly two 6's.
%Y A104729 Sequence in context: A015086 A052466 A078535 this_sequence A106661 A003720 A002884
%Y A104729 Adjacent sequences: A104726 A104727 A104728 this_sequence A104730 A104731 A104732
%K A104729 base,nonn
%O A104729 1,1
%A A104729 Shyam Sunder Gupta (guptass(AT)rediffmail.com), Apr 24 2005
%I A106661
%S A106661 6,166,4666,46666,1466666,5666666,116666666,166666666,13666666666,
%T A106661 46666666666,766666666666,20666666666666,106666666666666,
%U A106661 766666666666666,2666666666666666,46666666666666666,4066666666666666666
%N A106661 Smallest semiprime ending in exactly n 6's.
%e A106661 a(3)=4666 is a term because 4666 is the smallest semiprime ending in exactly three 6's.
%Y A106661 Sequence in context: A052466 A078535 A104729 this_sequence A003720 A002884 A055165
%Y A106661 Adjacent sequences: A106658 A106659 A106660 this_sequence A106662 A106663 A106664
%K A106661 base,nonn
%O A106661 1,1
%A A106661 Shyam Sunder Gupta (guptass(AT)rediffmail.com), May 12 2005
%I A003720 M4301
%S A003720 1,6,168,10672,1198080,208521728,51874413568,17449541107712,
%T A003720 7622674735988736,4193561606973095936,2836052065377836597248,
%U A003720 2312174256451088534208512,2236165580390456719589769216
%N A003720 Expansion of tan(tan(tan(x))).
%t A003720 Tan[ Tan[ Tan[ x ] ]] (* Odd Part *)
%Y A003720 Sequence in context: A078535 A104729 A106661 this_sequence A002884 A055165 A071095
%Y A003720 Adjacent sequences: A003717 A003718 A003719 this_sequence A003721 A003722 A003723
%K A003720 nonn
%O A003720 0,2
%A A003720 R. H. Hardin (rhh(AT)cadence.com)
%E A003720 Extended and formatted Mar 15 1997 by Olivier Gerard
%I A002884 M4302 N1798
%S A002884 1,1,6,168,20160,9999360,20158709760,163849992929280,5348063769211699200,
%T A002884 699612310033197642547200,366440137299948128422802227200,
%U A002884 768105432118265670534631586896281600
%N A002884 Number of nonsingular n X n matrices over GF(2) (order of Chevalley group A_n (2)).
%C A002884 Also (apparently) number of n X n matrices over GF(2) having permanent = 1. - Hugo Pfoertner (hugo(AT) pfoertner.org), Nov 14 2003. This is true because over GF(2) permanents and determinants are the same! - Joerg Arndt (arndt(AT)jjj.de), Mar 07 2008
%D A002884 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985, p. xvi.
%D A002884 H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.
%D A002884 P. F. Duvall, Jr., and P. W. Harley, III, A note on counting matrices, SIAM J. Appl. Math., 20 (1971), 374-377.
%D A002884 Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%D A002884 I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
%H A002884 T. D. Noe, Table of n, a(n) for n=0..30
%H A002884 J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher
%H A002884 Index entries for sequences related to binary matrices
%F A002884 Product(2^n-2^i, i=0..n-1); or 2^(n*(n-1)/2) * product( 2^i - 1, i=1..n).
%p A002884 product(2^n-2^i,i=0..n-1); or 2^(n*(n-1)/2) * product( 2^i - 1, i=1..n);
%Y A002884 Cf. A000409, A000410, A002820, A046747, A048651.
%Y A002884 Sequence in context: A104729 A106661 A003720 this_sequence A055165 A071095 A134632
%Y A002884 Adjacent sequences: A002881 A002882 A002883 this_sequence A002885 A002886 A002887
%K A002884 nonn,easy,nice
%O A002884 0,3
%A A002884 njas
%I A055165
%S A055165 1,6,174,22560,12514320,28836612000,270345669985440,
%T A055165 10160459763342013440
%N A055165 Number of regular n X n matrices with rational entries equal to 0 or 1.
%C A055165 All eigenvalues are nonzero.
%D A055165 Miodrag Zivkovic, Classification of small (0,1) matrices, Linear Algebra and its Applications, 414 (2006), 310-346
%H A055165 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%H A055165 Miodrag Zivkovic, More information
%H A055165 Index entries for sequences related to binary matrices
%F A055165 For an asymptotic estimate see A046747. A002884 is a lower bound. A002416 is an upper bound.
%F A055165 a(n) = n! * A088389(n) - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 20 2007
%e A055165 For n=2 the 6 matrices are {{{0, 1}, {1, 0}}, {{0, 1}, {1, 1}}, {{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}, {{1, 1}, {1, 0}}}.
%Y A055165 Cf. A056990, A056989, A046747, A055165, A002416, A003024 (positive definite matrices).
%Y A055165 A046747(n) + a(n) = 2^(n^2) = total number of n X n (0, 1) matrices = sequence A002416.
%Y A055165 Sequence in context: A106661 A003720 A002884 this_sequence A071095 A134632 A024277
%Y A055165 Adjacent sequences: A055162 A055163 A055164 this_sequence A055166 A055167 A055168
%K A055165 nonn,nice,hard
%O A055165 1,2
%A A055165 Ulrich Hermisson (uhermiss(AT)server1.rz.uni-leipzig.de), Jun 18 2000
%E A055165 More terms from MIodrag Zivkovic (ezivkovm(AT)matf.bg.ac.yu), Feb 28 2006
%I A071095
%S A071095 1,6,175,24696,16818516,55197331332,872299918503728,66345156372852988800,
%T A071095 24277282058281388285162560,42730166102274086598901662210000,361690697335823816369045433734882109375,
%U A071095 14721491647169381835282394824891766183125000000,2880942480871157389699990094736740229925045312500000000
%N A071095 Number of ways to tile hexagon of edges n, n+1, n+1, n, n+1, n+1 with diamonds of side 1.
%D A071095 J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see page 261).
%H A071095 J. Propp, Updated article
%H A071095 J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
%F A071095 Product_{i=0..a-1} Product_{j=0..b-1} Product_{k=0..c-1} (i+j+k+2)/(i+j+k+1) with a=n, b=c=n+1.
%Y A071095 Sequence in context: A003720 A002884 A055165 this_sequence A134632 A024277 A012177
%Y A071095 Adjacent sequences: A071092 A071093 A071094 this_sequence A071096 A071097 A071098
%K A071095 nonn
%O A071095 0,2
%A A071095 njas, May 28 2002
%I A134632
%S A134632 0,6,176,1278,5280,15950,39456,84966,165248,297270,502800,809006,
%T A134632 1249056,1862718,2696960,3806550,5254656,7113446,9464688
%N A134632 5*n^5 + 3*n^3 - 2*n^2. Coefficients and exponents are the prime numbers in decreasing order.
%F A134632 a(n) = 5*n^5 + 3*n^3 - 2*n^2.
%F A134632 G.f.: 2x*(3+70x+156x^2+66x^3+5x^4)/(1-x)^6. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 14 2007
%e A134632 a(4)=5280 because 4^5=1024, 5*1024=5120, 4^3=64, 3*64=192, 4^2=16, 2*16=32 and we can write 5120+192-32=5280.
%Y A134632 Cf. A000290, A000578, A000584, A045991, A100019, A133072.
%Y A134632 Sequence in context: A002884 A055165 A071095 this_sequence A024277 A012177 A012227
%Y A134632 Adjacent sequences: A134629 A134630 A134631 this_sequence A134633 A134634 A134635
%K A134632 nonn
%O A134632 0,2
%A A134632 Omar E. Pol (info(AT)polprimos.com), Nov 04 2007
%I A024277
%S A024277 0,1,6,176,8176,691456,86186496,15358324736,3667315849216,
%T A024277 1135407181398016,441731548179726336,211079248633366839296,
%U A024277 121507103129359646457856,82940335057202543199256576
%V A024277 0,1,-6,176,-8176,691456,-86186496,15358324736,-3667315849216,
%W A024277 1135407181398016,-441731548179726336,211079248633366839296,
%X A024277 -121507103129359646457856,82940335057202543199256576
%N A024277 Expansion of ln(1+tanh(x)*tan(x))/2.
%t A024277 Log[ 1+Tanh[ x ]*Tan[ x ]]/2 (* Even Part *)
%Y A024277 A009398.
%Y A024277 Cf. A101921.
%Y A024277 Sequence in context: A055165 A071095 A134632 this_sequence A012177 A012227 A012152
%Y A024277 Adjacent sequences: A024274 A024275 A024276 this_sequence A024278 A024279 A024280
%K A024277 sign
%O A024277 0,3
%A A024277 R. H. Hardin (rhh(AT)cadence.com)
%E A024277 Extended with signs 03/97.
%I A012177
%S A012177 1,6,176,11792,1399808,258010112,68048472064,24288413734912,
%T A012177 11265869660880896,6584336654436794368,4732528038730841194496,
%U A012177 4101916559962554300891136,4218542156655644369974460416
%N A012177 tan(tan(arctanh(x)))=x+6/3!*x^3+176/5!*x^5+11792/7!*x^7...
%Y A012177 Sequence in context: A071095 A134632 A024277 this_sequence A012227 A012152 A024278
%Y A012177 Adjacent sequences: A012174 A012175 A012176 this_sequence A012178 A012179 A012180
%K A012177 nonn
%O A012177 0,2
%A A012177 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A012227
%S A012227 1,6,176,12016,1469696,280971776,77198422016,28793654081536,
%T A012227 13988424620507136,8578076577450295296,6478101995566379565056,
%U A012227 5906079803365219349037056,6394716233788355978540875776
%V A012227 1,-6,176,-12016,1469696,-280971776,77198422016,-28793654081536,
%W A012227 13988424620507136,-8578076577450295296,6478101995566379565056,
%X A012227 -5906079803365219349037056,6394716233788355978540875776
%N A012227 tanh(arctan(tanh(x)))=x-6/3!*x^3+176/5!*x^5-12016/7!*x^7...
%Y A012227 Cf. A101921.
%Y A012227 Sequence in context: A134632 A024277 A012177 this_sequence A012152 A024278 A062240
%Y A012227 Adjacent sequences: A012224 A012225 A012226 this_sequence A012228 A012229 A012230
%K A012227 sign
%O A012227 0,2
%A A012227 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A012152
%S A012152 1,6,176,12240,1561088,318007808,94807748608,38942692972544,
%T A012152 21088169713729536,14558790290337431552,12481351528273319821312,
%U A012152 13009249833253480856289280,16200881782506627598590672896
%N A012152 arctanh(tan(tan(x)))=x+6/3!*x^3+176/5!*x^5+12240/7!*x^7...
%Y A012152 Sequence in context: A024277 A012177 A012227 this_sequence A024278 A062240 A046989
%Y A012152 Adjacent sequences: A012149 A012150 A012151 this_sequence A012153 A012154 A012155
%K A012152 nonn
%O A012152 0,2
%A A012152 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A024278
%S A024278 0,1,6,179,9676,854597,111361298,20061390071,4771665341848,
%T A024278 1447947783210249,545795035419716382,250167844738073041595,
%U A024278 137013989756844496168292,88367083240335992790622797
%N A024278 Expansion of tan(tan(x))*sin(x)/2.
%t A024278 Tan[ Tan[ x ]]*Sin[ x ]/2 (* Even Part *)
%Y A024278 A009698.
%Y A024278 Sequence in context: A012177 A012227 A012152 this_sequence A062240 A046989 A135395
%Y A024278 Adjacent sequences: A024275 A024276 A024277 this_sequence A024279 A024280 A024281
%K A024278 nonn
%O A024278 0,3
%A A024278 R. H. Hardin (rhh(AT)cadence.com)
%E A024278 Extended and signs tested 03/97.
%I A062240
%S A062240 1,6,179,48337
%N A062240 Number of subgroups of Chevalley group A_n(2) (the group of nonsingular n X n matrices over GF(2) ).
%Y A062240 A002884.
%Y A062240 Sequence in context: A012227 A012152 A024278 this_sequence A046989 A135395 A141121
%Y A062240 Adjacent sequences: A062237 A062238 A062239 this_sequence A062241 A062242 A062243
%K A062240 nonn,hard
%O A062240 1,2
%A A062240 Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 30 2001
%I A046989
%S A046989 1,6,180,2835,37800,467775,3831077250,127702575,2605132530000,350813659321125,
%T A046989 15313294652906250,147926426347074375,2423034863565078262500,144228265688397515625,
%U A046989 3952575621190533915703125,84913182070036240111050234375,999843529136357459316262500000
%N A046989 Denominators of Taylor series expansion of log(x/sin x).
%D A046989 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 222, series for log(H(x)/x).
%D A046989 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A046989 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
%e A046989 log(x/sin(x)) = 1/6*x^2+1/180*x^4+1/2835*x^6+1/37800*x^8+1/467775*x^10+...
%Y A046989 Cf. A046988.
%Y A046989 Sequence in context: A012152 A024278 A062240 this_sequence A135395 A141121 A051357
%Y A046989 Adjacent sequences: A046986 A046987 A046988 this_sequence A046990 A046991 A046992
%K A046989 nonn,easy,frac,nice
%O A046989 0,2
%A A046989 njas
%I A135395
%S A135395 6,180,5040,143640,4199580,125621496,3830266440,118655943120,
%T A135395 3724872182460,118248726796200,3789926661961440,122473276342326000,
%U A135395 3986235855826497000,130561182081992667600,4300094066688571550400
%N A135395 Number of walks from origin to (1,1,1) on a cubic lattice.
%C A135395 a(n) is the number of walks of length 2n+3 in a cubic lattice that begin at the origin and end at (1,1,1) using steps (1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1).
%H A135395 S. Hollos and R. Hollos, Lattice Paths and Walks.
%F A135395 a(n) = binomial(2n+3,n) * sum( binomial(n,k) * binomial(n+3,k+2) * binomial(2k+2,k+1), k, 0, n )
%o A135395 Maxima: a(n) = binomial(2n+3,n) * sum( binomial(n,k) * binomial(n+3,k+2) * binomial(2k+2,k+1), k, 0, n )
%Y A135395 Cf. A002896.
%Y A135395 Sequence in context: A024278 A062240 A046989 this_sequence A141121 A051357 A064120
%Y A135395 Adjacent sequences: A135392 A135393 A135394 this_sequence A135396 A135397 A135398
%K A135395 easy,nonn
%O A135395 0,1
%A A135395 Stefan Hollos (stefan(AT)exstrom.com), Dec 11 2007
%I A141121
%S A141121 1,6,180,8640,498960,31434480,2055943296,135216506304,8720972739072,
%T A141121 538646016002688,31024094144060160,1609593032459782656,
%U A141121 71392972690228672512,2461961564459510280192,51302015299696881770496
%V A141121 1,6,-180,8640,-498960,31434480,-2055943296,135216506304,-8720972739072,
%W A141121 538646016002688,-31024094144060160,1609593032459782656,-71392972690228672512,
%X A141121 2461961564459510280192,-51302015299696881770496,-415041229811424576835584
%N A141121 G.f. A(x) satisfies: A(A(A(A(A(A(x)))))) = x + 36*x^2.
%e A141121 G.f.: A(x) = x + 6*x^2 - 180*x^3 + 8640*x^4 - 498960*x^5 +...
%e A141121 A(A(x)) = x + 12*x^2 - 288*x^3 + 12096*x^4 - 622080*x^5 +...
%e A141121 A(A(A(x))) = x + 18*x^2 - 324*x^3 + 11664*x^4 - 524880*x^5 +...
%e A141121 A(A(A(A(x)))) = x + 24*x^2 - 288*x^3 + 8640*x^4 - 331776*x^5 +...
%e A141121 A(A(A(A(A(x))))) = x + 30*x^2 - 180*x^3 + 4320*x^4 - 136080*x^5 +...
%o A141121 (PARI) {a(n, m=6)=local(F=x+m*x^2+x*O(x^n), G); if(n<1, 0, for(k=3, n, G=F+x*O(x^k); for(i=1, m-1, G=subst(F, x, G)); F=F+((-polcoeff(G, k))/m)*x^k); return(polcoeff(F, n, x)))}
%Y A141121 Cf. A027436, A141119, A141119, A141120.
%Y A141121 Sequence in context: A062240 A046989 A135395 this_sequence A051357 A064120 A041429
%Y A141121 Adjacent sequences: A141118 A141119 A141120 this_sequence A141122 A141123 A141124
%K A141121 sign
%O A141121 1,2
%A A141121 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 05 2008
%I A051357
%S A051357 1,6,180,37800,87318000,2622159540000,1338638666765400000,
%T A051357 12984380089637682726000000,2896722619368127899492763620000000,
%U A051357 18740906719713843949122453226304292600000000
%N A051357 Chernoff sequence A006939 divided by 2.
%F A051357 a(n) = Product_{k=1..n} prime(k)^(n-k+1) / 2
%Y A051357 Sequence in context: A046989 A135395 A141121 this_sequence A064120 A041429 A089905
%Y A051357 Adjacent sequences: A051354 A051355 A051356 this_sequence A051358 A051359 A051360
%K A051357 easy,nonn
%O A051357 1,2
%A A051357 Judson D. Neer (judson(AT)poboxes.com)
%I A064120
%S A064120 1,6,180,873600,772107033600
%N A064120 A036981(n)/n!^2.
%H A064120 Index entries for sequences related to tournaments
%Y A064120 Sequence in context: A135395 A141121 A051357 this_sequence A041429 A089905 A012208
%Y A064120 Adjacent sequences: A064117 A064118 A064119 this_sequence A064121 A064122 A064123
%K A064120 nonn,hard
%O A064120 0,2
%A A064120 njas, Dec 01 2001
%I A041429
%S A041429 1,6,181,1092,32941,198738,5995081,36169224,1091071801,
%T A041429 6582600030,198569072701,1197997036236,36138480159781,218028877994922,
%U A041429 6577004820007441,39680057798039568,1196978738761194481
%N A041429 Denominators of continued fraction convergents to sqrt(230).
%Y A041429 Cf. A041428.
%Y A041429 Sequence in context: A141121 A051357 A064120 this_sequence A089905 A012208 A012181
%Y A041429 Adjacent sequences: A041426 A041427 A041428 this_sequence A041430 A041431 A041432
%K A041429 nonn,cofr,easy
%O A041429 0,2
%A A041429 njas
%I A089905
%S A089905 0,6,183,3285,47787,627789,7777791,92777793,1077777795,12277777797,
%T A089905 137777777799,1527777777801,16777777777803,182777777777805,
%U A089905 1977777777777807,21277777777777809,227777777777777811
%N A089905 Sum of digits of numbers between 0 and (3/9)*(10^n-1).
%C A089905 From a suggestion of Yalcin Aktar (aktaryalcin(AT)msn.com)
%F A089905 a(n)=s(3, n-1) where s(a, k)=a*(k+1)+a^2*sum(i=0, k, i*10^(k-i))+sum(i=0, k, 5*a*(9*(k-i)+a- 1)*10^(k-i-1))
%F A089905 Terms satisfy a(n)=22a(n-1)-141a(n-2)+220a(n-3)-100a(n-4) - T. D. Noe (noe(AT)sspectra.com), Nov 08 2006
%Y A089905 Cf. A089903, A089904, A089906, A087330, A089906, A089907, A089908, A034967.
%Y A089905 Sequence in context: A051357 A064120 A041429 this_sequence A012208 A012181 A012224
%Y A089905 Adjacent sequences: A089902 A089903 A089904 this_sequence A089906 A089907 A089908
%K A089905 nonn
%O A089905 0,2
%A A089905 Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 14 2003
%E A089905 Corrected by T. D. Noe (noe(AT)sspectra.com), Nov 08 2006
%I A012208
%S A012208 1,6,184,13136,1679488,335723264,96492653568,37665610561536,
%T A012208 19158060131778560,12304193595906129920,9734505561510299566080,
%U A012208 9299736312230986726768640,10553185490408007072659537920
%V A012208 1,-6,184,-13136,1679488,-335723264,96492653568,-37665610561536,
%W A012208 19158060131778560,-12304193595906129920,9734505561510299566080,
%X A012208 -9299736312230986726768640,10553185490408007072659537920
%N A012208 tanh(arctan(atan(x)))=x-6/3!*x^3+184/5!*x^5-13136/7!*x^7...
%Y A012208 Sequence in context: A064120 A041429 A089905 this_sequence A012181 A012224 A037298
%Y A012208 Adjacent sequences: A012205 A012206 A012207 this_sequence A012209 A012210 A012211
%K A012208 sign
%O A012208 0,2
%A A012208 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A012181
%S A012181 1,6,184,13360,1770880,373587200,115086003200,48784879769600,
%T A012181 27245388132352000,19392357120188416000,17137521906875269120000,
%U A012181 18411376349575875461120000,23632217190341837269237760000
%N A012181 arctanh(tan(atanh(x)))=x+6/3!*x^3+184/5!*x^5+13360/7!*x^7...
%Y A012181 Sequence in context: A041429 A089905 A012208 this_sequence A012224 A037298 A015004
%Y A012181 Adjacent sequences: A012178 A012179 A012180 this_sequence A012182 A012183 A012184
%K A012181 nonn
%O A012181 0,2
%A A012181 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A012224
%S A012224 1,6,184,13584,1840768,397140224,124961670144,54024939321344,
%T A012224 30732245464219648,22258174840910315520,20000813079319781048320,
%U A012224 21837489688957057049821184,28476242253811580631861166080
%V A012224 1,-6,184,-13584,1840768,-397140224,124961670144,-54024939321344,
%W A012224 30732245464219648,-22258174840910315520,20000813079319781048320,
%X A012224 -21837489688957057049821184,28476242253811580631861166080
%N A012224 arctan(atan(tanh(x)))=x-6/3!*x^3+184/5!*x^5-13584/7!*x^7...
%Y A012224 Sequence in context: A089905 A012208 A012181 this_sequence A037298 A015004 A129046
%Y A012224 Adjacent sequences: A012221 A012222 A012223 this_sequence A012225 A012226 A012227
%K A012224 sign
%O A012224 0,2
%A A012224 Patrick Demichel (dml(AT)hpfrcu03.france.hp.com)
%I A037298
%S A037298 6,186,6877,222943,6084393,154793510
%N A037298 Number of (s,6) gates.
%D A037298 E Detjens and G Gannot, Technology mapping in MIS, pp. 116-119 of some 1987 IEEE Conference Proceedings [ # CH2469-5/87 ].
%Y A037298 Sequence in context: A012208 A012181 A012224 this_sequence A015004 A129046 A059491
%Y A037298 Adjacent sequences: A037295 A037296 A037297 this_sequence A037299 A037300 A037301
%K A037298 nonn
%O A037298 1,1
%A A037298 njas
%I A015004
%S A015004 1,6,186,29016,22661496,88515803376,1728802155736656,
%T A015004 168827903320618878336,82435457461295106532780416,
%U A015004 201258420458750640859769304304896
%N A015004 q-factorial numbers for q=5.
%H A015004 Index entries for sequences related to factorial numbers
%F A015004 prod_{k=1}^{k=n} {(q^k - 1) / (q - 1)}
%Y A015004 Sequence in context: A012181 A012224 A037298 this_sequence A129046 A059491 A024279
%Y A015004 Adjacent sequences: A015001 A015002 A015003 this_sequence A015005 A015006 A015007
%K A015004 nonn,easy
%O A015004 1,2
%A A015004 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A129046
%S A129046 0,6,187,4462,86968,1448516,21535942,294625589,3787236314,
%T A129046 46411139226,547841825257
%N A129046 Number of n-node triangulations of the Klein bottle N_2 in which every node has degree >= 3.
%D A129046 G. Ringel, Wie man die geschlossenen nichtorientierbaren Flaechen in moeglichst wenig Dreiecke zerlegen kann, Math. Ann. 130 (1955), 317-326.
%H A129046 Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).
%Y A129046 Sequence in context: A012224 A037298 A015004 this_sequence A059491 A024279 A012205
%Y A129046 Adjacent sequences: A129043 A129044 A129045 this_sequence A129047 A129048 A129049
%K A129046 nonn
%O A129046 7,2
%A A129046 njas, May 13 2007
%I A059491
%S A059491 1,1,6,189,30618,25332021,106698472452,2283997201168644,248218139523497121576,
%T A059491 136861610819571430116630660,382684747771430768732371981946100,
%U A059491 5424628155237728987530088501811168904125,389729317367139375014273384868937660572301897500
%N A059491 Expansion of generating function A_{QT}^(1)(4n;3).
%H A059491 G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv math.CO/0008184 [Th. 5]
%F A059491 (3^(n*(n-1)/2)*A005130(n).
%F A059491 a(n+1) is the Hankel transform of A097188. Odd terms occur in a(n+1) at positions given by 2*A000975(n). - Paul Barry (pbarry(AT)wit.ie), Feb 09 2007
%Y A059491 Sequence in context: A037298 A015004 A129046 this_sequence A024279 A012205 A086065
%Y A059491 Adjacent sequences: A059488 A059489 A059490 this_sequence A059492 A059493 A059494
%K A059491 nonn,easy
%O A059491 0,3
%A A059491 njas, Feb 04 2001
%I A024279
%S A024279 0,1,6,191,11676,1111501,159118146,31534949291,8253261002616,
%T A024279 2758076443359961,1145356666834820286,578481632054752663511,
%U A024279 349162848547458172567956,248194708790049332718453541
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