A255611 G.f.: Product_{k>=1} 1/(1-x^k)^(4*k).

1, 4, 18, 64, 215, 660, 1938, 5400, 14527, 37728, 95278, 234344, 563506, 1326796, 3066040, 6963048, 15564661, 34282360, 74486376, 159785472, 338703796, 709957616, 1472529670, 3023894672, 6151408852, 12402137024, 24792822174, 49162962280, 96737562642

Offset

0,2

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

Formula

G.f.: Product_{k>=1} 1/(1-x^k)^(4*k).

a(n) ~ 2^(1/3) * Zeta(3)^(5/18) * exp(1/3 + 3 * Zeta(3)^(1/3) * n^(2/3)) / (A^4 * sqrt(3*Pi) * n^(7/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015

G.f.: exp(4*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

Maple

a:= proc(n) option remember; `if`(n=0, 1, 4*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)^(4*k), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000219, A161870, A255610, A255612, A255613, A255614, A193427.

Column k=4 of A255961.

Sequence in context: A100177 A083321 A362316 * A022728 A231950 A246134

Adjacent sequences: A255608 A255609 A255610 * A255612 A255613 A255614

Keyword

nonn

Author

Vaclav Kotesovec, Feb 28 2015

Extensions

New name from Vaclav Kotesovec, Mar 12 2015

Status

approved