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A263143 Expansion of Product_{k>=1} 1/(1-x^(5*k-3))^k.

%I A263143 #9 Oct 12 2015 10:31:02
%S A263143 1,0,1,0,1,0,1,2,1,2,1,2,4,2,7,2,7,6,7,12,7,16,12,16,26,16,35,22,40,
%T A263143 44,40,68,47,80,84,86,133,94,167,146,182,238,198,314,274,358,431,386,
%U A263143 593,494,696,754,761,1056,917,1288,1326,1430,1880,1664,2348,2290
%N A263143 Expansion of Product_{k>=1} 1/(1-x^(5*k-3))^k.
%H A263143 Vaclav Kotesovec, <a href="/A263143/b263143.txt">Table of n, a(n) for n = 0..10000</a>
%H A263143 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 A263143 G.f.: exp(Sum_{j>=1} 1/j*x^(2*j)/(1 - x^(5*j))^2).
%F A263143 a(n) ~ Zeta(3)^(121/900) * exp(d53 - Pi^4/(1200*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (100*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(329/900) * 5^(121/450) * sqrt(3*Pi) * n^(571/900)), where d53 = A263180 = Integral_{x=0..infinity} exp(-2*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 3/(25*x^2) - 29/(300*x*exp(x)) = -0.1461681349208040073620067065149936790708... .
%p A263143 with(numtheory):
%p A263143 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263143       `if`(irem(d+5, 5, 'r')=2, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263143     end:
%p A263143 seq(a(n), n=0..100); # after _Alois P. Heinz_
%t A263143 nmax = 100; CoefficientList[Series[Product[1/(1-x^(5k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263143 nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(2*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263143 Cf. A263141, A263142, A263144, A263180, A263147.
%K A263143 nonn
%O A263143 0,8
%A A263143 _Vaclav Kotesovec_, Oct 10 2015