%I A000108 M1459 N0577
%S A000108 1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,
%T A000108 2674440,9694845,35357670,129644790,477638700,1767263190,
%U A000108 6564120420,24466267020,91482563640,343059613650,1289904147324
%N A000108 Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also
called Segner numbers.
%C A000108 The solution to Schroeder's first problem. A very large number of combinatorial
interpretations are known - see references, esp. Stanley, Enumerative
Combinatorics, Volume 2.
%C A000108 Number of ways to insert n pairs of parentheses in a word of n+1 letters.
E.g. for n=3 there are 5 ways: ((ab)(cd)), (((ab)c)d), ((a(bc))d),
(a((bc)d)), (a(b(cd))).
%C A000108 Consider all the binomial(2n,n) paths on squared paper that (i) start
at (0, 0), (ii) end at (2n, 0) and (iii) at each step, either make
a (+1,+1) step or a (+1,-1) step. Then the number of such paths which
never go never below the x-axis is C(n) [Chung-Feller]
%C A000108 a(n) is the number of ordered rooted trees with n nodes, not including
the root. See the Conway-Guy reference where these rooted ordered
trees are called plane bushes. See also the Bergeron et al. reference,
Example 4, p. 167. W. Lang Aug 07 2007.
%C A000108 Shifts one place left when convolved with itself.
%C A000108 For n >= 1 a(n) is also the number of rooted bicolored unicellular maps
of genus 0 on n edges. - Ahmed Fares (ahmedfares(AT)my-deja.com),
Aug 15 2001
%C A000108 Ways of joining 2n points on a circle to form n nonintersecting chords.
(If no such restriction imposed, then ways of forming n chords is
given by (2n-1)!!=(2n)!/n!2^n=A001147(n).)
%C A000108 Arises in Schubert calculus - see Sottile reference.
%C A000108 Inverse Euler transform of sequence is A022553.
%C A000108 With interpolated zeros, the inverse binomial transform of the Motzkin
numbers A001006. - Paul Barry (pbarry(AT)wit.ie), Jul 18 2003
%C A000108 The Hankel transforms of this sequence or of this sequence with the first
term omitted give A000012 = 1, 1, 1, 1, 1, 1, ...; example : Det([1,
1, 2, 5; 1, 2, 5, 14; 2, 5, 14, 42; 5, 14, 42, 132]) = 1 and Det([1,
2, 5, 14; 2, 5, 14, 42; 5, 14, 42, 132; 14, 42, 132, 429]) = 1 .
- DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 04 2004
%C A000108 c(n) = C(2*n-2,n-1)/n = (1/n!) * [ n^(n-1) + { C(n-2,1) +C(n-2,2) }*n^(n-2)
+ { 2*C(n-3,1) +7*C(n-3,2) +8*C(n-3,3) +3*C(n-3,4) }*n^(n-3) + {
6*C(n-4,1) +38*C(n-4,2) +93*C(n-4,3) +111*C(n-4,4) +65*C(n-4,5) +15*C(n-4,
6) }*n^(n-4) + ..... ]. - Andre F. Labossiere (boronali(AT)laposte.net),
Nov 10 2004
%C A000108 Sum_{n=0..infinity} 1/a(n) = 2 + 4*Pi/3^(5/2) = F(1,2;1/2;1/4) = 2.806133050770763...
(see L'Universe de Pi link) - Gerald McGarvey and Benoit Cloitre,
Feb 13 2005
%C A000108 a(n) equals sum of squares of terms in row n of triangle A053121, which
is formed from successive self-convolutions of the Catalan sequence.
- Paul D. Hanna (pauldhanna(AT)juno.com), Apr 23 2005
%C A000108 Comment from Donald D. Cross (cosinekitty(AT)hotmail.com), Feb 04 2005:
Also coefficients of the Mandelbrot polynomial M iterated an infinite
number of times. Examples: M(0) = 0 = 0*c^0 = [0], M(1) = c = c^1
+ 0*c^0 = [1 0], M(2) = c^2 + c = c^2 + c^1 + 0*c^0 = [1 1 0], M(3)
= (c^2 + c)^2 + c = [0 1 1 2 1], ... ... M(5) = [0 1 1 2 5 14 26
44 69 94 114 116 94 60 28 8 1], ...
%C A000108 The multiplicity with which a prime p divides C_n can be determined by
first expressing n+1 in base p. For p=2, the multiplicity is the
number of 1 digits minus 1. For p an odd prime, count all digits
greater than (p+1)/2; also count digits equal to (p+1)/2 unless final;
and count digits equal to (p-1)/2 if not final and the next digit
is counted. For example, n=62, n+1 = 223_5, so C_62 is not divisible
by 5. n=63, n+1 = 224_5, so 5^3 | C_63. - Frank Adams-Watters (FrankTAW(AT)Netscape.net),
Feb 08 2006
%C A000108 Koshy and Salmassi give an elementary proof that the only prime Catalan
numbers are a(2) = 2 and a(3) = 5. Is the only semiprime Catalan
number a(4) = 14? - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar
06 2006
%C A000108 Comment from Franklin T. Adams-Watters, Apr 14 2006: The answer is yes.
Using the formula C_n = C(2n,n)/(n+1), it is immediately clear that
C_n can have no prime factor greater than 2n. For n >= 7, C_n > (2n)^2,
so it cannot be a semiprime. Given that the Catalan numbers grow
exponentially, the above consideration implies that the number of
prime divisors of C_n, counted with multiplicity, must grow without
limit. The number of distinct prime divisors must also grow without
limit, but this is more difficult. Any prime between n+1 and 2n (exclusive)
must divide C_n. That the number of such primes grows without limit
follows from the prime number theorem.
%C A000108 The number of ways to place n indistinguishable balls in n numbered boxes
B1,...,Bn such that at most a total of k balls are placed in boxes
B1,...,Bk for k=1,...,n. For example, a(3)=5 since there are 5 ways
to distribute 3 balls among 3 boxes such that (i) box 1 gets at most
1 ball and (ii)box 1 and box 2 together get at most 2 balls:(O)(O)(O),
(O)()(OO), ()(OO)(O), ()(O)(OO), ()()(OOO). - Dennis P. Walsh (dwalsh(AT)mtsu.edu),
Dec 04 2006
%C A000108 a(n) is also the order of the semigroup of order-decreasing and order-preserving
full transformations (of an n-element chain) - now known as the Catalan
monoid [From A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008]
%C A000108 a(n) is the number of trivial representations in the direct product of
2n spinor (the smallest) representations of the group SU(2) (A(1)).
[From Rutger Boels (boels(AT)nbi.dk), Aug 26 2008]
%C A000108 The invert transform appears to converge to the catalan numbers when
applied infinitely many times to any starting sequence. [From Mats
O. Granvik, Gary W. Adamson and Roger L. Bagula (mgranvik(AT)abo.fi),
Sep 09 2008, Sep 12 2008]
%C A000108 lim(a(n)/a(n-1): n->infinity) = 4 [From Francesco Antoni (francesco_antoni(AT)yahoo.com),
Nov 24 2008]
%C A000108 Starting with offset 1 = row sums of triangle A154559 [From Gary W. Adamson
(qntmpkt(AT)yahoo.com), Jan 11 2009]
%C A000108 Contribution from Benji Fisher (benji(AT)FisherFam.org), Mar 05 2009:
(Start)
%C A000108 C(n) is the degree of the Grassmanian G(1,n+1): the set of lines in
%C A000108 (n+1)-dimensional projective space, or the set of planes through the
origin in
%C A000108 (n+2)-dimensional affine space. The Grassmanian is considered a subset
of
%C A000108 N-dimensional projective space, N = binomial(n+2,2) - 1. If we choose
2n
%C A000108 general (n-1)-planes in projective (n+1)-space, then there are C(n) lines
that
%C A000108 meet all of them. (End)
%C A000108 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2009:
(Start)
%C A000108 Starting with offset 1 = A068875: (1, 2, 4, 10, 18, 84,...) convolved
with
%C A000108 Fine numbers, A000957: (1, 0, 1, 2, 6, 18,...). a(6) = 132 =
%C A000108 (1, 2, 4, 10, 28, 84) dot (18, 6, 2, 1, 0, 1) = (18 + 12 + 8 + 10 + 0
+ 84) = 132. (End)
%C A000108 Convolved with A032443: (1, 3, 11, 42, 163,...) = powers of 4, A000302:
(1, 4, 16,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May
15 2009]
%C A000108 Sum{k=1...Infinity,c(k-1)/2^(2k-1)}=1. The kth term in the summation
is the probability that a random walk on the integers (begining at
the origin) will arrive at positive one (for the first time) in exactly
(2k-1) steps. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org),
Sep 12 2009]
%D A000108 The large number of references and links demonstrates the ubiquity of
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%H A000108 John Baez, <a href="http://math.ucr.edu/home/baez/week202.html">This
week's finds in mathematical physics, Week 202</a>
%H A000108 E. Barcucci, A. Del Lungo, E. Pergola and R. Pinzani, <a href="http:/
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avoiding an increasing number of length-increasing forbidden subsequences</
a>
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0205301">Some canonical sequences of integers</a>, Linear Alg. Applications,
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%H A000108 D. Bill, <a href="http://www.durangobill.com/BinTrees.html">Durango Bill's
Enumeration of Binary Trees</a>
%H A000108 H. Bottomley, <a href="a108b.gif">Catalan Space Invaders</a>
%H A000108 H. Bottomley, <a href="a002694.gif">Illustration for A000108, A001147,
A002694, A067310 and A067311</a>
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montagnards.pdf">Montagnards et polygones</a>
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The Meanings of Catalan Numbers</a>
%H A000108 B. Bukh, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
CatalanNumbers.html">Catalan numbers</a>
%H A000108 Alexander Burstein, Sergi Elizalde and Toufik Mansour, <a href="http:/
/arXiv.org/abs/math.CO/0610234">Restricted Dumont permutations, Dyck
paths and noncrossing partitions</a>, arXiv math.CO/0610234.
%H A000108 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">
Random walks, trees and extensions of Riordan group techniques</a>
%H A000108 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000108 F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/
NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies
in Automatic Combinatorics, Volume II (1997).
%H A000108 Julie Christophe, Jean-Paul Doignon and Samuel Fiorini, <a href="http:/
/www.cs.uwaterloo.ca/journals/JIS/index.html">Counting Biorders</
a>, J. Integer Seqs., Vol. 6, 2003.
%H A000108 T. Davis, <a href="http://www.geometer.org/mathcircles/catalan.pdf">Catalan
Numbers</a>
%H A000108 E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/pdf/math.CO/0407326">
Congruences for Catalan and Motzkin numbers and related sequences</
a>, J. Num. Theory 117 (2006), 191-215.
%H A000108 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/catalan.html">
Catalan numbers</a>
%H A000108 R. M. Dickau, <a href="http://www-groups.dcs.st-andrews.ac.uk/~history/
Miscellaneous/CatalanNumbers/catalan.html">Catalan Numbers (another
copy)</a>
%H A000108 D. Foata and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/
mamarim/mamarimPDF/classic.pdf">A classic proof of a recurrence for
a very classical sequence</a>
%H A000108 I. Galkin, <a href="http://ulcar.uml.edu/~iag/CS/Catalan.html">Enumeration
of the Binary Trees(Catalan Numbers)</a>
%H A000108 H. W. Gould, <a href="http://www.ams.org/mathscinet-getitem?mr=2049119">
Congr. Numer. 165 (2003) p 33-38</a>.
%H A000108 B. Gourevitch, <a href="http://www.pi314.net/">L'univers de Pi</a> (click
Mathematiciens, Gosper)
%H A000108 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</
a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=48">
Encyclopedia of Combinatorial Structures 48</a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=52">
Encyclopedia of Combinatorial Structures 52</a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=71">
Encyclopedia of Combinatorial Structures 71</a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=76">
Encyclopedia of Combinatorial Structures 76</a>
%H A000108 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=284">
Encyclopedia of Combinatorial Structures 284</a>
%H A000108 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/polygons/Polygons_ser.html">
Series exapansions for self-avoiding polygons</a>
%H A000108 S. Johnson, <a href="http://www.saintanns.k12.ny.us/depart/math/Seth/
catafrm.html">The Catalan Numbers</a>
%H A000108 A. Karttunen, <a href="http://ndirty.cute.fi/%7Ekarttu/matikka/Nekomorphisms/
a014486.ps.gz">Illustration of initial terms up to size n=7</a>
%H A000108 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Matrix Transformations of Integer Sequences</a>, J. Integer Seqs.,
Vol. 6, 2003.
%H A000108 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
Sobalian Coefficients</a>.
%H A000108 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
index.html">Miscellaneous</a>.
%H A000108 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
On generalizations of Stirling number triangles</a>, J. Integer Seqs.,
Vol. 3 (2000), #00.2.4.
%H A000108 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
The Hankel Transform and Some of its Properties</a>, J. Integer Sequences,
4 (2001), #01.1.5.
%H A000108 Colin L. Mallows and Lou Shapiro, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Balls on the Lawn</a>, J. Integer Sequences,
Vol. 2, 1999, #5.
%H A000108 Toufik Mansour, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Counting Peaks at Height k in a Dyck Path</a>, Journal of Integer
Sequences, Vol. 5 (2002), Article 02.1.1
%H A000108 R. J. Marsh and P. P. Martin, <a href="http://arXiv.org/abs/math.CO/0612572">
Pascal arrays: counting Catalan sets</a>
%H A000108 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/
~merlini/fun01.ps">Waiting patterns for a printer</a>, FUN with algorithm'01,
Isola d'Elba, 2001.
%H A000108 A. Panayotopoulos and P. Tsikouras, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Meanders and Motzkin Words</a>, J. Integer
Seqs., Vol. 7, 2004.
%H A000108 A. Panholzer and H. Prodinger, <a href="http://www.wits.ac.za/helmut/
abstract/abs_159.htm">Bijections for ternary trees and non-crossing
trees</a>, Discrete Math., 250 (2002), 181-195 (see Eq. 4).
%H A000108 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/
JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</
a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A000108 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/
JIS/index.html">Dyck Paths With No Peaks at Height k</a>, J. Integer
Sequences, 4 (2001), #01.1.3.
%H A000108 K. A. Penson and J.-M. Sixdeniers, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Integral Representations of Catalan and
Related Numbers</a>, J. Integer Sequences, 4 (2001), #01.2.5.
%H A000108 A. Sapounakis and P. Tsikouras, <a href="http://www.cs.uwaterloo.ca/journals/
JIS/">On k-colored Motzkin words</a>, Journal of Integer Sequences,
Vol. 7 (2004), Article 04.2.5.
%H A000108 N. J. A. Sloane, <a href="a108.html">Illustration of initial terms</a>
%H A000108 Frank Sottile, <a href="http://www.math.umass.edu/~sottile/pages/ERAG/
S4/1.html">The Schubert Calculus of Lines</a> (a section of Enumerative
Real Algebraic Geometry)
%H A000108 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html">Hipparchus,
Plutarch, Schr"oder and Hough</a>, Am. Math. Monthly, Vol. 104, No.
4, p. 344, 1997.
%H A000108 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec/catalan.pdf">
Exercises on Catalan and Related Numbers</a>
%H A000108 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec#catadd">Catalan
Addendum</a>
%H A000108 Zhi-Wei Sun, <a href="http://front.math.ucdavis.edu/math.CO/0509648">
A combinatorial identity with application to Catalan numbers</a>
%H A000108 D. Taylor, <a href="http://www.maths.usyd.edu.au/u/don/code/Catalan/Catalan.html">
Catalan Structures(up to C(7))</a>
%H A000108 G. Villemin's Almanac of Numbers, <a href="http://perso.wanadoo.fr/yoda.guillaume/
Nb30a50/Catalan.htm">Nombres De Catalan</a>
%H A000108 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CatalanNumber.html">Link to a section of The World of Mathematics
(1).</a>
%H A000108 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BinaryBracketing.html">Link to a section of The World of Mathematics
(2).</a>
%H A000108 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BinaryTree.html">Link to a section of The World of Mathematics (3).</
a>
%H A000108 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
NonassociativeProduct.html">Link to a section of The World of Mathematics
(4).</a>
%H A000108 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
StaircaseWalk.html">Link to a section of The World of Mathematics
(5).</a>
%H A000108 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
DyckPath.html">Dyck Path</a>
%H A000108 W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Hankel Matrices and Lattice Paths</a>, J. Integer Sequences, 4 (2001),
#01.1.2.
%H A000108 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000108 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to
rooted trees</a>
%H A000108 <a href="Sindx_Par.html#parens">Index entries for sequences related to
parenthesizing</a>
%H A000108 <a href="Sindx_Ne.html#necklaces">Index entries for sequences related
to necklaces</a>
%H A000108 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
Publications/books.html">Analytic Combinatorics</a>, 2009; see page
18, 35
%F A000108 a(n) = binomial(2n, n)/(n+1) = (2n)!/(n!(n+1)!).
%F A000108 a(n) = binomial(2n, n)-binomial(2n, n-1)
%F A000108 a(n) = Sum_{k=0..n-1} a(k)a(n-1-k).
%F A000108 G.f.: A(x) = (1 - sqrt(1 - 4*x)) / (2*x). G.f. A(x) satisfies A = 1 +
x*A^2.
%F A000108 a(n+1) = Sum_{i} binomial(n, 2*i)*2^(n-2*i)*a(i) - Touchard.
%F A000108 2(2n-1)a(n-1) = (n+1)a(n).
%F A000108 It is known that a(n) is odd if and only if n=2^k-1, k=1, 2, 3, ... -
Emeric Deutsch, Aug 04 2002.
%F A000108 Using the Stirling approximation in A000142 we get the asymptotic expansion
a(n) ~ 4^n / (sqrt(Pi * n) * (n + 1)). - Dan Fux (dan.fux(AT)OpenGaia.com
or danfux(AT)OpenGaia.com), Apr 13 2001
%F A000108 Integral representation: a(n)=int(x^n*sqrt((4-x)/x), x=0..4)/(2*Pi).
- Karol A. Penson (penson(AT)lptl.jussieu.fr), Apr 12 2001
%F A000108 E.g.f.: exp(2x) (I_0(2x)-I_1(2x)), where I_n is Bessel function. - Karol
A. Penson (penson(AT)lptl.jussieu.fr), Oct 07 2001
%F A000108 Polygorial(n, 6)/Polygorial(n, 3) - Daniel Dockery (peritus(AT)gmail.com)
Jun 24, 2003
%F A000108 G.f. A(x) satisfies ((A(x)+A(-x))/2)^2 = A(4*x^2). - Michael Somos, Jun
27, 2003
%F A000108 G.f. A(x) satisfies Sum_{k>=1} k(A(x)-1)^k = Sum_{n >= 1} 4^{n-1} x^n.
- Shapiro, Woan, Getu
%F A000108 a(n+m) = Sum_{k} A039599(n, k)*A039599(m, k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Dec 22 2003
%F A000108 a(n+1) = (1/(n+1))*sum_{k=0..n} a(n-k)*binomial(2k+1, k+1) . - DELEHAM
Philippe (kolotoko(AT)wanadoo.fr), Jan 24 2004
%F A000108 a(n) = Sum_{k>=0} A008313(n, k)^2 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Feb 14 2004
%F A000108 a(m+n+1) = Sum_{k>=0} A039598(m, k)*A039598(n, k) . - DELEHAM Philippe
(kolotoko(AT)wanadoo.fr), Feb 15 2004
%F A000108 a(n)=sum{k=0..n, (-1)^k*2^(n-k)*binomial(n, k)*binomial(k, floor(k/2))}
- Paul Barry (pbarry(AT)wit.ie), Jan 27 2005
%F A000108 a(n) = Sum_{k=0..[n/2]} ((n-2*k+1)*C(n, n-k)/(n-k+1))^2, which is equivalent
to: a(n) = Sum_{k=0..n} A053121(n, k)^2, for n>=0. - Paul D. Hanna
(pauldhanna(AT)juno.com), Apr 23 2005
%F A000108 a((m+n)/2) = Sum_{k>=0} A053121(m, k)*A053121(n, k) if m+n is even .
- Philippe DELEHAM, May 26 2005
%F A000108 E.g.f. Sum_{n>=0} a(n)*x^(2n)/(2n)! = BesselI(1, 2x)/x . - Michael Somos
Jun 22 2005
%F A000108 Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(x, B(X)) where f(u,
v)=u-v+(uv)^2 or B(x)=x+(x*B(x))^2 which implies B(-B(x))=-x and
also (1+B^3)/B^2 = (1-x^3)/x^2 . - Michael Somos Jun 27 2005
%F A000108 a(n) = a(n-1)*(4-6/(n+1)). a(n) = 2a(n-1)*(8a(n-2)+a(n-1))/(10a(n-2)-a(n-1)).
- Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 08 2006
%F A000108 Sum_{k=1}^{infinity} a(k)/4^k = 1. - Frank Adams-Watters (FrankTAW(AT)Netscape.net),
Jun 28 2006
%F A000108 a(n) = A047996(2*n+1,n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Jul 25 2006
%F A000108 Binomial transform of A005043 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Oct 20 2006
%F A000108 a(n)=Sum_{k, 0<=k<=n}(-1)^k*A116395(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Nov 07 2006
%F A000108 a(n)=[1/(s-n)]*sum_{k=0..n} (-1)^k (k+s-n)*binomial(s-n,k)*binomial(s+n-k,
s) with s a nonnegative free integer [H. W. Gould].
%F A000108 a(k) = Sum_{i=1..k} |A008276(i,k)| * (k-1)^(k-i) / k! - Andre F. Labossiere
(boronali(AT)laposte.net), May 29 2007
%F A000108 a(n)=Sum_{k, 0<=k<=n}A129818(n,k)*A007852(k+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Jun 20 2007
%F A000108 a(n)=Sum_{k, 0<=k<=n}A109466(n,k)*A127632(k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Jun 20 2007
%F A000108 Row sums of triangle A124926 - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Oct 22 2007
%F A000108 For G.f. A(x), g(x)= x*A(x) is the compositional inverse of f(x) = x*(1-x)
and this relates the Catalan numbers to the row sums of A125181.
- Tom Copeland (tcjpn(AT)msn.com), Jan 13 2008
%F A000108 lim(1+Sum(a(k)/A004171(k): 0<=k<=n): n->infinity) = 4/pi. [From Reinhard
Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 26 2008]
%F A000108 a(n)=Sum_{k, 0<=k<=n}A120730(n,k)^2 and a(k+1)=Sum_{n, n>=k}A120730(n,
k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2008]
%F A000108 Comment from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 27 2008: Given
an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}],
we may define an infinite sequence Phi(u) by setting a_n = a_{n-1}
+ a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example
the present sequence is Phi([1]) (also Phi([1,1])).
%F A000108 Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009:
%F A000108 a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}
%F A000108 delta(l_1,l_2,...,l_i,...,l_n)
%F A000108 where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i < l_(i+1) and l_(i+1)
<> 0
%F A000108 for i=1..n-1 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.
%F A000108 a(n) = C(2*n,n)-C(2*n,n-1) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 17 2009]
%p A000108 A000108 := n->binomial(2*n,n)/(n+1); G000108 := (1 - sqrt(1 - 4*x)) /
(2*x);
%p A000108 spec := [ A, {A=Prod(Z,Sequence(A))}, unlabeled ]: [ seq(combstruct[count](spec,
size=n), n=0..42) ];
%p A000108 with(combstruct):bin := {B=Union(Z,Prod(B,B))}: seq (count([B,bin,unlabeled],
size=n), n=1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Dec 05 2007
%p A000108 Z[0]:=0: for k to 42 do Z[k]:=simplify(1/(1-z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]),
j=1..42): gser:=series(g, z=0, 42): seq(coeff(gser, z, n), n=0..41);
- Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 21 2008
%t A000108 A000108[ n_ ] := (2 n)!/n!/(n+1)!
%t A000108 A000108[n_]:=Hypergeometric2F1[1-n,-n,2,1] - Richard L. Ollerton (r.ollerton(AT)uws.edu.au),
Sep 13 2006
%o A000108 (MAGMA) C:= func< n | Binomial(2*n,n)/(n+1) >; [ C(n) : n in [0..60]];
%o A000108 (PARI) a(n)=if(n<0,0,(2*n)!/n!/(n+1)!)
%o A000108 (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=1+x+O(x^2); while(m<=n,m*=2;
A=sqrt(subst(A,x,4*x^2)); A+=(A-1)/(2*x*A)); polcoeff(A,n))
%o A000108 (PARI) a(n)=if(n<1,n==0,polcoeff(serreverse(x/(1+x)^2+x*O(x^n)),n)) (from
Michael Somos)
%o A000108 (Mupad) combinat::dyckWords::count(n) $ n = 0..38 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 14 2007
%o A000108 sage: [catalan_number(i) for i in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 26 2008
%o A000108 (Other) sage: [binomial(2*i,i)-binomial(2*i,i-1) for i in xrange(0,25)]#
[From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
%Y A000108 Cf. A000984, A002420, A048990, A024492, A000142, A022553. A row of A060854.
%Y A000108 Cf. A039599, A094216, A094638, A014137, A094639, A099731, A008549.
%Y A000108 See A001003, A001190, A001699, A000081 for other ways to count parentheses.
%Y A000108 Enumerates objects encoded by A014486.
%Y A000108 Cf. A008276, A094638 (|A008276|), A094216, A094639, A000984.
%Y A000108 A diagonal of any of the essentially equivalent arrays A009766, A030237,
A033184, A059365, A099039, A106566, A130020, A047072.
%Y A000108 Cf. A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392.
%Y A000108 Cf. A124926.
%Y A000108 Cf. A098597, A086117, A137697.
%Y A000108 A diagonal of the square array described in A051168.
%Y A000108 A154559 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 11 2009]
%Y A000108 A000957, A068875 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01
2009]
%Y A000108 A032443 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
%Y A000108 Sequence in context: A054394 A036769 A033191 this_sequence A115140 A120588
A057413
%Y A000108 Adjacent sequences: A000105 A000106 A000107 this_sequence A000109 A000110
A000111
%K A000108 core,nonn,easy,eigen,nice
%O A000108 0,3
%A A000108 N. J. A. Sloane (njas(AT)research.att.com).
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