cp's OEIS Frontend

A263141 Expansion of Product_{k>=1} 1/(1-x^(5*k-1))^k.

%I A263141 #10 Oct 12 2015 10:29:56
%S A263141 1,0,0,0,1,0,0,0,1,2,0,0,1,2,3,0,1,2,6,4,1,2,6,10,6,2,6,14,20,8,6,14,
%T A263141 29,30,13,14,34,54,50,22,34,66,99,74,43,72,133,166,119,82,148,242,276,
%U A263141 182,166,286,438,442,301,316,541,744,701,494,608,976,1255
%N A263141 Expansion of Product_{k>=1} 1/(1-x^(5*k-1))^k.
%H A263141 Vaclav Kotesovec, <a href="/A263141/b263141.txt">Table of n, a(n) for n = 0..10000</a>
%H A263141 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 A263141 G.f.: exp(Sum_{j>=1} 1/j*x^(4*j)/(1 - x^(5*j))^2).
%F A263141 a(n) ~ Zeta(3)^(169/900) * exp(d51 - Pi^4/(10800*Zeta(3))+ Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (300 * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(281/900) * 5^(169/450) * sqrt(3*Pi) * n^(619/900)), where d51 = A263178 = Integral_{x=0..infinity} exp(-4*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 1/(25*x^2) + 19/(300*x*exp(x)) = -0.1269958671388232529452705747311358056... .
%p A263141 with(numtheory):
%p A263141 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263141       `if`(irem(d+5, 5, 'r')=4, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263141     end:
%p A263141 seq(a(n), n=0..100); # after _Alois P. Heinz_
%t A263141 nmax = 100; CoefficientList[Series[Product[1/(1-x^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263141 nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(4*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263141 Cf. A263142, A263143, A263144, A263178, A263145, A035528, A262876, A263136.
%K A263141 nonn
%O A263141 0,10
%A A263141 _Vaclav Kotesovec_, Oct 10 2015