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A263137 Expansion of Product_{k>=1} 1/(1-x^(4*k-3))^k.

%I A263137 #11 Jan 07 2021 07:32:07
%S A263137 1,1,1,1,1,3,3,3,3,6,9,9,9,13,19,23,23,28,42,51,56,62,84,108,120,133,
%T A263137 170,219,253,276,335,427,503,556,650,815,977,1090,1244,1525,1836,2079,
%U A263137 2344,2808,3386,3876,4348,5107,6121,7069,7932,9176,10918,12671,14257
%N A263137 Expansion of Product_{k>=1} 1/(1-x^(4*k-3))^k.
%H A263137 Vaclav Kotesovec, <a href="/A263137/b263137.txt">Table of n, a(n) for n = 0..10000</a>
%H A263137 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 A263137 G.f.: exp(Sum_{j>=1} 1/j*x^j/(1 - x^(4*j))^2).
%F A263137 a(n) ~ Zeta(3)^(29/288) * exp(d43 - Pi^4/(768*Zeta(3)) + Pi^2 * n^(1/3) / (16*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3)/4) / (2^(77/96) * sqrt(3*Pi) * n^(173/288)), where d43 = A263177 = Integral_{x=0..infinity} exp(-x)/(x*(1 - exp(-4*x))^2) - 1/(16*x^3) - 3/(16*x^2) - 19/(96*x*exp(x)) dx = 0.0960101036186695795680588847641594939129540181270663556962564550198... .
%p A263137 with(numtheory):
%p A263137 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263137       `if`(irem(d+4, 4, 'r')=1, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263137     end:
%p A263137 seq(a(n), n=0..100); # after _Alois P. Heinz_
%t A263137 nmax = 100; CoefficientList[Series[Product[1/(1-x^(4k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263137 nmax = 100; CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263137 Cf. A263136, A263143, A263177, A263139.
%K A263137 nonn
%O A263137 0,6
%A A263137 _Vaclav Kotesovec_, Oct 10 2015