2,2

Ways of placing n labeled balls into 2 indistinguishable boxes with at least 2 balls in each box.

2^a(n) = integer values of the form 1/(2-sum(i=1,m, i/2^i)). - Benoit Cloitre Oct 25 2002

Number of permutations avoiding 13-2 that contain the pattern 23-1 exactly twice.

Cost of ternary maximum height Huffman tree with N internal nodes (non-leaves) for minimizing absolutely ordered sequences of size n=2N+1. - Alex Vinokur (alexvn(AT)barak-online.net), Nov 02 2004

a(n)=number of Dyck (n+3)-paths whose third upstep initiates the last long ascent, n>=1. A long ascent is one consisting of 2 or more upsteps. For example, a(1)=3 counts UUDuUDDD, UDUDuUDD, UUDDuUDD (third upstep in small type). - David Callan (callan(AT)stat.wisc.edu), Dec 08 2004

A107907(a(n)) = A000225(n+2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 28 2005

No exponentiation is needed. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 10 2008]

Subsequence of A158581; A000120(a(n)) > 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 16 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Amer. Math. Monthly, Vol. 101 (No. 8, Oct 1994), p. 776.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.

F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

T. D. Noe, Table of n, a(n) for n=2..300

T. Mansour, Restricted permutations by patterns of type 2-1.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Alex Vinokur, Fibonacci-like polynomials produced by m-ary Huffman codes for absolutely ordered sequences, E-print

E.g.f.: ((exp(x)-1-x)^2)/2!. G.f.: (3-2*x)/((1-2*x)*(1-x)^2)

a(n) = 2*a(n-1)+n+3 = a(n)+2^(n+2)-1 = A000295(n+3)-1 = A000295(n+4)-2^(n+3).

Starting (3, 10, 25, 56,...) = binomial transform of [3, 7, 8, 8, 8,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 07 2007

a(4)=4!/(2!*2!*2!)=3

with(combinat):a:=n->sum(sum(binomial(j, k), j=2..n), k=1..n): seq(a(n), n=1..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 10 2007

A000247:=(-3+2*z)/((2*z-1)*(z-1)**2); [S. Plouffe in his 1992 dissertation.]

a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=n+2*a[n-1]+1 od: seq(a[n], n=1..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008

restart:with (combinat):a:=n->add(stirling2(j, 2), j=3..n): seq(a(n), n=2..25); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 01 2009]

lst={}; s=1; Do[s+=(s-n); AppendTo[lst, Abs[s]], {n, 2, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 10 2008]

Table[Sum[Binomial[n, i - 1], {i, 3, n}], {n, 2, 41}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]

(Other) sage: [gaussian_binomial(n, 1, 2)-(n+1) for n in xrange(2, 32)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009]

Cf. A000478 (3 boxes), A058844 (4 boxes).

Sequence in context: A053208 A162607 A047667 this_sequence A097763 A034506 A067988

Adjacent sequences: A000244 A000245 A000246 this_sequence A000248 A000249 A000250

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Michael Steyer (msteyer(AT)osram.de), Dec 02 2000. More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000

I recently changed the beginning of this sequence so the formulae etc. may need to be adjusted.