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%I #8 Oct 12 2015 10:28:51
%S 1,1,0,0,0,2,2,0,0,3,4,1,0,4,10,6,0,5,16,14,3,6,28,32,10,7,40,63,33,
%T 11,60,112,74,23,80,187,161,56,111,300,308,131,152,455,568,295,223,
%U 672,968,607,356,967,1609,1186,618,1367,2546,2189,1132,1926,3941
%N Expansion of Product_{k>=1} (1+x^(4*k-3))^k.
%H Vaclav Kotesovec, <a href="/A263139/b263139.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^j/(1 - x^(4*j))^2).
%F a(n) ~ 2^(83/96) * 3^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(2304*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(2/3) * n^(1/3) / (96*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 2^(-8/3) * 3^(4/3) * n^(2/3)) / (12 * sqrt(Pi) * n^(2/3)).
%t nmax = 100; CoefficientList[Series[Product[(1+x^(4k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^j/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A263138, A263137, A263147.
%K nonn
%O 0,6
%A _Vaclav Kotesovec_, Oct 10 2015
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