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A263138 Expansion of Product_{k>=1} (1+x^(4*k-1))^k.

%I A263138 #8 Oct 12 2015 10:28:22
%S A263138 1,0,0,1,0,0,0,2,0,0,2,3,0,0,4,4,0,1,10,5,0,6,16,6,0,14,28,7,3,32,40,
%T A263138 8,10,63,60,9,33,112,80,13,74,187,110,25,161,300,140,58,308,455,183,
%U A263138 133,568,672,236,297,968,963,321,609,1609,1344,468,1188,2546
%N A263138 Expansion of Product_{k>=1} (1+x^(4*k-1))^k.
%H A263138 Vaclav Kotesovec, <a href="/A263138/b263138.txt">Table of n, a(n) for n = 0..10000</a>
%H A263138 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
%F A263138 G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(3*j)/(1 - x^(4*j))^2).
%F A263138 a(n) ~ 2^(59/96) * 3^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(20736*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(2/3) * n^(1/3) / (288*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 2^(-8/3) * 3^(4/3) * n^(2/3)) / (12 * sqrt(Pi) * n^(2/3)).
%t A263138 nmax = 100; CoefficientList[Series[Product[(1+x^(4k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263138 nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(3*j)/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263138 Cf. A263139, A263136, A263140, A262878, A263145.
%K A263138 nonn
%O A263138 0,8
%A A263138 _Vaclav Kotesovec_, Oct 10 2015