A005986 Number of column-strict plane partitions of n. (Formerly M1393)

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

0,2

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.

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

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 *)

Prog

(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

Crossrefs

Cf. A003293.

Sequence in context: A227637 A171985 A354044 * A333396 A277828 A147878

Adjacent sequences: A005983 A005984 A005985 * A005987 A005988 A005989

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, May 21 2006

Status

approved