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A263147 Expansion of Product_{k>=1} (1+x^(5*k-3))^k.

%I A263147 #8 Oct 12 2015 10:33:08
%S A263147 1,0,1,0,0,0,0,2,0,2,0,0,3,0,4,0,1,4,0,10,0,6,5,0,16,0,14,6,3,28,0,32,
%T A263147 7,10,40,0,63,8,33,60,3,112,9,74,80,14,187,10,161,110,46,300,12,308,
%U A263147 140,120,455,24,568,182,283,672,54,968,224,594,963,146
%N A263147 Expansion of Product_{k>=1} (1+x^(5*k-3))^k.
%H A263147 Vaclav Kotesovec, <a href="/A263147/b263147.txt">Table of n, a(n) for n = 0..10000</a>
%H A263147 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 A263147 G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(2*j)/(1 - x^(5*j))^2).
%F A263147 a(n) ~ 2^(43/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(3600*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (300*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 3^(4/3) * 2^(2/3) * 5^(1/3) * n^(2/3) / 20) / (30 * sqrt(Pi) * n^(2/3)).
%t A263147 nmax = 100; CoefficientList[Series[Product[(1+x^(5k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263147 nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(2*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263147 Cf. A263145, A263146, A263148, A263143.
%K A263147 nonn
%O A263147 0,8
%A A263147 _Vaclav Kotesovec_, Oct 10 2015