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A255610 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k).

%I A255610 #34 Feb 16 2025 08:33:25
%S A255610 1,3,12,37,111,303,804,2022,4950,11715,27081,61083,135112,293142,
%T A255610 625620,1314267,2722323,5564172,11234865,22424904,44284545,86573147,
%U A255610 167648418,321746907,612274678,1155782109,2165116416,4026391221,7435806048,13641093684,24865920932
%N A255610 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k).
%H A255610 Seiichi Manyama, <a href="/A255610/b255610.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vaclav Kotesovec)
%H A255610 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
%H A255610 Vaclav Kotesovec, <a href="/A255610/a255610.jpg">Graph - The asymptotic ratio (250000 terms)</a>
%H A255610 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanePartition.html">Plane Partition</a>
%H A255610 Wikipedia, <a href="http://en.wikipedia.org/wiki/Plane_partition">Plane partition</a>
%F A255610 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k).
%F A255610 a(n) ~ Zeta(3)^(1/4) * exp(1/4 + 2^(-2/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^3 * 6^(1/4) * sqrt(Pi) * n^(3/4)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - _Vaclav Kotesovec_, Feb 28 2015
%F A255610 More precise asymptotics: a(n) ~ Zeta(3)^(1/4) * exp(1/4 + 2^(-2/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^3 * 6^(1/4) * sqrt(Pi) * n^(3/4)) * (1 - c/n^(2/3)), where c = 0.21774822... . - _Vaclav Kotesovec_, Oct 15 2015
%F A255610 G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - _Ilya Gutkovskiy_, May 29 2018
%p A255610 a:= proc(n) option remember; `if`(n=0, 1, 3*add(
%p A255610       a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
%p A255610     end:
%p A255610 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 11 2015
%t A255610 nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(3*k),{k,1,nmax}],{x,0,nmax}],x]
%Y A255610 Cf. A000219, A161870, A255611, A255612, A255613, A255614, A193427.
%Y A255610 Column k=3 of A255961.
%K A255610 nonn
%O A255610 0,2
%A A255610 _Vaclav Kotesovec_, Feb 28 2015
%E A255610 New name from _Vaclav Kotesovec_, Mar 12 2015