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%I #12 Oct 15 2015 10:39:20
%S 1,1,1,1,1,1,3,3,3,3,3,6,9,9,9,9,13,19,23,23,23,28,42,51,56,56,62,84,
%T 108,120,126,133,170,219,253,268,283,335,427,503,547,574,658,815,977,
%U 1080,1144,1265,1534,1836,2068,2209,2408,2832,3396,3864,4178,4505
%N Expansion of Product_{k>=1} 1/(1-x^(5*k-4))^k.
%C In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} 1/(1 - x^(s*k-t))^k then a(n) ~ s^(t^2/(3*s^2) - 7/18) * n^(t^2/(6*s^2) - 25/36) * exp(d(s,t) - Pi^4 * t^2 / (432*s^2 * Zeta(3)) + Pi^2 * t * 2^(2/3) * s^(2/3) * n^(1/3) / (12 * s^2 * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / (2^(2/3)*s^(2/3))) / (2^(t^2/(6*s^2) + 11/36) * sqrt(3*Pi) * Zeta(3)^(t^2/(6*s^2) - 7/36)), where d(s,t) = Integral_{x=0..infinity} 1/x * (exp(-(s-t)*x)/(1 - exp(-s*x))^2 - 1/(s^2*x^2) - t/(s^2*x) + exp(-x)*(1/12 - t^2/(2*s^2))) dx.
%H Vaclav Kotesovec, <a href="/A263144/b263144.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 G.f.: exp(Sum_{j>=1} 1/j*x^j/(1 - x^(5*j))^2).
%F a(n) ~ Zeta(3)^(79/900) * exp(d54 - Pi^4/(675*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (75*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(371/900) * 5^(79/450) * sqrt(3*Pi) * n^(529/900)), where d54 = A263181 = Integral_{x=0..infinity} exp(-x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 4/(25*x^2) - 71/(300*x*exp(x)) = 0.1863826906247526303913683646299184833844240863417644... .
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p `if`(irem(d+5, 5, 'r')=1, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..100); # after _Alois P. Heinz_
%t nmax = 100; CoefficientList[Series[Product[1/(1-x^(5k-4))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 100; CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A263141, A263142, A263143, A263181, A263148.
%K nonn
%O 0,7
%A _Vaclav Kotesovec_, Oct 10 2015
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