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A263149 Expansion of Product_{k>=1} (1 + x^(2*k+1))^k.

%I A263149 #8 Oct 12 2015 10:34:28
%S A263149 1,0,0,1,0,2,0,3,2,4,4,5,10,7,16,13,28,22,40,41,63,73,90,123,143,199,
%T A263149 214,316,343,483,532,733,848,1099,1305,1644,2029,2448,3067,3657,4643,
%U A263149 5443,6892,8107,10224,12031,14974,17798,21941,26190,31867,38381,46300
%N A263149 Expansion of Product_{k>=1} (1 + x^(2*k+1))^k.
%H A263149 Vaclav Kotesovec, <a href="/A263149/b263149.txt">Table of n, a(n) for n = 0..10000</a>
%H A263149 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 A263149 G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(3*j)/(1 - x^(2*j))^2).
%F A263149 a(n) ~ exp(-Pi^4/(5184*Zeta(3)) - Pi^2 * n^(1/3) / (8 * 3^(4/3) * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)/4) * Zeta(3)^(1/6) / (2^(23/24) * 3^(1/3)* sqrt(Pi) * n^(2/3)).
%t A263149 nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k+1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263149 nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(3*j)/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263149 Cf. A035528, A263140, A263150, A263199.
%K A263149 nonn
%O A263149 0,6
%A A263149 _Vaclav Kotesovec_, Oct 10 2015