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%I #7 Apr 25 2017 04:55:57
%S 1,1,9,36,148,489,1959,6326,22741,74072,246436,781189,2523042,7773342,
%T 24200874,73439472,222247101,660405663,1958564056,5715567301,
%U 16623600991,47780474694,136623175876,386983158080,1090779014163,3048348195528,8478106666045
%N Expansion of Product_{k>=1} 1/(1 - k^2*x^k)^k.
%H Vaclav Kotesovec, <a href="/A285674/b285674.txt">Table of n, a(n) for n = 0..3000</a>
%F a(n) ~ c * 3^(2*n/3) * n^2, where
%F c = 76631915822.1860553820452485980060616094557062528483009... if mod(n,3)=0
%F c = 76631915822.1819974623120987784506295282600132985390786... if mod(n,3)=1
%F c = 76631915822.1825610530012010285873110459423711856434442... if mod(n,3)=2
%F In closed form, a(n) ~ (Product_{k>=4}((1 - k^2/3^(2*k/3))^(-k)) / ((1 - 1/3^(2/3)) * (1 - 4/3^(4/3))^2) + Product_{k>=4}((1 - (-1)^(2*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(2*n/3) * ((1 + 4/3*(-1/3)^(1/3))^2 * (1 - (-1/3)^(2/3)))) + Product_{k>=4}((1 - (-1)^(4*k/3)*k^2/3^(2*k/3))^(-k)) / ((-1)^(4*n/3) * ((1 + (-1)^(1/3)/3^(2/3)) * (1 - 4*(-1)^(2/3) / 3^(4/3))^2))) * 3^(2*n/3) * n^2 / 54.
%t nmax=40; CoefficientList[Series[Product[1/(1-k^2*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A077335, A266941, A285241, A285737.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Apr 24 2017
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