A005986 Number of column-strict plane partitions of n.

1, 2, 5, 11, 23, 45, 87, 160, 290, 512, 889, 1514, 2547, 4218, 6909, 11184, 17926, 28449, 44772, 69862, 108205, 166371, 254107, 385617, 581729, 872535, 1301722, 1932006, 2853530, 4194867, 6139361, 8946742, 12984724, 18771092, 27033892

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Note that the asymptotic formula by Gordon and Houten, cited in Stanley's paper (proposition 20.3, p. 274) is for sequence A003293, not for A005986. In addition in the same paper proposition 20.2 is wrong and Wright's formula is incomplete (for correct version see A000219). - Vaclav Kotesovec, Feb 28 2015

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Graph - The asymptotic ratio.
• Richard P. Stanley, Theory and Application of Plane Partitions, II, Studies in Appl. Math. 50 (1971), 259-279. DOI:10.1002/sapm1971503259.

Crossrefs

Cf. A003293.

Programs

Maple

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n fi end end: a:=etr(n-> `if`(modp(n, 2)=0, n+2, n+3)/2): seq(a(n), n=0..45);  # Vaclav Kotesovec, Mar 02 2015 after Alois P. Heinz

Mathematica

CoefficientList[ Series[ Product[1/((1 - x^i)*Product[(1 - x^j), {j, 2 i - 1, 40}]), {i, 40}], {x, 0, 40}], x] (* or *)
CoefficientList[ Series[ Product[1/(1 - x^j)^Floor[(j + 3)/2], {j, 40}], {x, 0, 40}], x] (* Robert G. Wilson v, May 12 2014 *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^((2*k+5-(-1)^k)/4),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

PARI

A005986_list(N,x=(O('x^N)+1)*'x)=Vec(prod(k=1,N,1/(1-x^k)^((k+3)\2))) \\ M. F. Hasler, Sep 26 2018

Formula

G.f.: 1/Product((1-x^i)*Product(1-x^j,j=2*i-1..infinity),i=1..infinity) or 1/Product((1-x^i)^floor((i+3)/2),i=1..infinity). - Vladeta Jovovic, May 21 2006

a(n) ~ Zeta(3)^(25/72) * exp(1/24 - 25*Pi^4 / (3456*Zeta(3)) + 5*Pi^2*n^(1/3) / (24*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3)*n^(2/3) / 2) / (A^(1/2) * 2^(5/4) * 3^(1/2) * Pi * n^(61/72)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 07 2015

Extensions

More terms from Vladeta Jovovic, May 21 2006