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A263142 Expansion of Product_{k>=1} 1/(1-x^(5*k-2))^k.

%I A263142 #9 Oct 12 2015 10:30:29
%S A263142 1,0,0,1,0,0,1,0,2,1,0,2,1,3,2,1,6,2,5,6,2,11,6,7,15,6,21,15,12,30,15,
%T A263142 34,35,22,58,35,59,70,43,108,76,95,142,85,187,157,161,263,174,318,307,
%U A263142 274,480,336,534,583,479,836,649,879,1068,840,1433,1211
%N A263142 Expansion of Product_{k>=1} 1/(1-x^(5*k-2))^k.
%H A263142 Vaclav Kotesovec, <a href="/A263142/b263142.txt">Table of n, a(n) for n = 0..10000</a>
%H A263142 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 A263142 G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(5*j))^2).
%F A263142 a(n) ~ Zeta(3)^(151/900) * exp(d52 - Pi^4/(2700*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (150 * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(299/900) * 5^(151/450) * sqrt(3*Pi) * n^(601/900)), where d52 = A263179 = Integral_{x=0..infinity} exp(-3*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 2/(25*x^2) + 1/(300*x*exp(x)) = -0.187803021063745858976409657887070138806... .
%p A263142 with(numtheory):
%p A263142 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263142       `if`(irem(d+5, 5, 'r')=3, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263142     end:
%p A263142 seq(a(n), n=0..100); # after _Alois P. Heinz_
%t A263142 nmax = 100; CoefficientList[Series[Product[1/(1-x^(5k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263142 nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263142 Cf. A263141, A263143, A263144, A263179, A263146, A262877.
%K A263142 nonn
%O A263142 0,9
%A A263142 _Vaclav Kotesovec_, Oct 10 2015