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A001523 Number of stacks, or planar partitions of n; also weakly unimodal partitions of n.
(Formerly M1102 N0420)
27
1, 1, 2, 4, 8, 15, 27, 47, 79, 130, 209, 330, 512, 784, 1183, 1765, 2604, 3804, 5504, 7898, 11240, 15880, 22277, 31048, 43003, 59220, 81098, 110484, 149769, 202070, 271404, 362974, 483439, 641368, 847681, 1116325, 1464999, 1916184, 2498258 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) counts stacks of integer-length boards of total length n where no board overhangs the board underneath.

Number of graphical partitions on 2n nodes that contain a 1. E.g. a(3)=4 and so there are 4 graphical partitions of 6 that contain a 1, namely (111111), (21111), (2211) and (3111). Only (222) fails. - Jon Perry, Jul 25 2003

It would seem from Stanley that he regards a(0)=0 for this sequence and A001522. - Michael Somos, Feb 22 2015

In the article by Auluck is a typo in the formula (24), 2*Pi is missing in an exponent on the left side of the equation for Q(n). The correct term is exp(2*Pi*sqrt(n/3)), not just exp(sqrt(n/3)). - Vaclav Kotesovec, Jun 22 2015

REFERENCES

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

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

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76.

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686, g(x).

F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)

H. Bottomley, Illustration of initial terms

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46

R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms

A. D. Sokal, The leading root of the partial theta function, arXiv preprint arXiv:1106.1003, 2011.

E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

FORMULA

a(n) = Sum(1 <= k <= n, f(k, n-k)), where f(n, k) (=A054250) = 1 if k = 0; Sum(1 <= j <= min(n, k); (n-j+1) f(j, k-j)) if k > 0.

a(n) = sum_k[A059623(n, k)] for n>0 - Henry Bottomley, Feb 01 2001

A006330(n) + a(n) = A000712(n). - Michael Somos, Jul 22 2003

G.f.: 1 + (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k))^2. - Michael Somos, Jul 22 2003

G.f.: 1 + sum(n>=1, x^n / ( prod(k=1..n-1, 1-x^k)^2 * (1-x^n) ) ). [Joerg Arndt, Oct 01 2012]

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015

EXAMPLE

For a(4)=8 we have the following stacks:

x

x x. .x

x x. .x x.. .x. ..x xx

x xx xx xxx xxx xxx xx xxxx

G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 15*x^5 + 27*x^6 + 47*x^7 + 79*x^8 + ...

MAPLE

b:= proc(n, i) option remember;

      `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+

      add(b(n-i*j, i+1)*(j+1), j=0..n/i))

    end:

a:= n-> `if`(n=0, 1, b(n, 1)):

seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

MATHEMATICA

max = 40; s = 1 + Sum[(-1)^(k + 1)*q^(k*(k + 1)/2), {k, 1, max}] / QPochhammer[q]^2 + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Jan 25 2012, updated Nov 29 2015 *)

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i]==0, 1, 0] + Sum[b[n-i*j, i+1]*(j+1), {j, 0, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 24 2015, after Alois P. Heinz *)

PROG

(PARI) {a(n) = if( n<1, n==0, polcoeff( sum(k=1, (sqrt(1 + 8*n) - 1)\2, -(-1)^k * x^((k + k^2)/2)) / eta(x + x * O(x^n))^2 , n))}; /* Michael Somos, Jul 22 2003 */

CROSSREFS

Cf. A054250, A059618, A059623, A001522, A001524.

Cf. A000569. Bisections give A100505, A100506.

Row sums of A247255.

Sequence in context: A054174 A239890 A222038 * A222039 A222148 A222040

Adjacent sequences:  A001520 A001521 A001522 * A001524 A001525 A001526

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Formula and more terms from David W. Wilson May 05 2000.

STATUS

approved

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Last modified March 25 00:06 EDT 2017. Contains 284035 sequences.