A255612 G.f.: Product_{k>=1} 1/(1-x^k)^(5*k).

1, 5, 25, 100, 370, 1251, 4005, 12150, 35400, 99365, 270353, 715025, 1844650, 4652075, 11494605, 27872056, 66428295, 155809600, 360079225, 820715820, 1846583863, 4104572975, 9019869125, 19608423750, 42193733645, 89917531549, 189863358445, 397401303850

Offset: 0

Author

Vaclav Kotesovec, Feb 28 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
• Eric Weisstein's World of Mathematics, Plane Partition
• Wikipedia, Plane partition

Crossrefs

Cf. A000219, A161870, A255610, A255611, A255613, A255614, A193427.

Column k=5 of A255961.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, 5*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(5*k),{k,1,nmax}],{x,0,nmax}],x]

Formula

G.f.: Product_{k>=1} 1/(1-x^k)^(5*k).

a(n) ~ 5^(11/36) * Zeta(3)^(11/36) * exp(5/12 + 3 * 2^(-2/3) * 5^(1/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^5 * 2^(7/36) * sqrt(3*Pi) * n^(29/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015

G.f.: exp(5*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

Extensions

New name from Vaclav Kotesovec, Mar 12 2015