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A263136 Expansion of Product_{k>=1} 1/(1-x^(4*k-1))^k.

%I A263136 #10 Oct 12 2015 10:26:59
%S A263136 1,0,0,1,0,0,1,2,0,1,2,3,1,2,6,5,2,6,11,7,6,15,21,12,15,30,34,22,35,
%T A263136 58,59,43,70,108,95,85,142,187,161,167,263,318,274,318,480,534,471,
%U A263136 595,836,879,819,1081,1433,1442,1429,1915,2391,2365,2483,3314,3947
%N A263136 Expansion of Product_{k>=1} 1/(1-x^(4*k-1))^k.
%H A263136 Vaclav Kotesovec, <a href="/A263136/b263136.txt">Table of n, a(n) for n = 0..10000</a>
%H A263136 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 A263136 G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(4*j))^2).
%F A263136 a(n) ~ Zeta(3)^(53/288) * exp(d41 - Pi^4/(6912*Zeta(3)) + Pi^2 * n^(1/3) / (48*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3)/4) / (sqrt(3*Pi) * 2^(101/96) * n^(197/288)), where d41 = A263176 = Integral_{x=0..infinity} exp(-3*x)/(x*(1 - exp(-4*x))^2) - 1/(16*x^3) - 1/(16*x^2) + 5/(96*x*exp(x)) = -0.158924147180165035059952001737321408554746599955833696821824808027... .
%p A263136 with(numtheory):
%p A263136 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263136       `if`(irem(d+4, 4, 'r')=3, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263136     end:
%p A263136 seq(a(n), n=0..100); # after _Alois P. Heinz_
%t A263136 nmax = 100; CoefficientList[Series[Product[1/(1-x^(4k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263136 nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263136 Cf. A035528, A262876, A263141, A263137, A263176, A263138.
%K A263136 nonn
%O A263136 0,8
%A A263136 _Vaclav Kotesovec_, Oct 10 2015