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A263395 Expansion of Product_{k>=1} 1/(1 - x^(2*k+5))^k.

%I A263395 #16 Nov 16 2024 16:32:52
%S A263395 1,0,0,0,0,0,0,1,0,2,0,3,0,4,1,5,2,6,6,7,10,9,19,11,28,16,44,25,61,40,
%T A263395 87,65,116,107,160,168,215,260,295,393,407,578,573,836,814,1193,1167,
%U A263395 1675,1684,2335,2427,3238,3501,4468,5014,6161,7152,8494,10121
%N A263395 Expansion of Product_{k>=1} 1/(1 - x^(2*k+5))^k.
%H A263395 Vaclav Kotesovec, <a href="/A263395/b263395.txt">Table of n, a(n) for n = 0..5000</a>
%H A263395 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 A263395 G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^(2*k))^2)).
%F A263395 a(n) ~ 2^(109/72) * exp(-1/24 - 25*Pi^4/(1728*Zeta(3)) - 5*Pi^2 * n^(1/3) / (12 * 2^(2/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(25/72) / (3*sqrt(3*Pi) * Zeta(3)^(61/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263395 with(numtheory):
%p A263395 a:= proc(n) option remember; local r; `if`(n=0, 1,
%p A263395        add(add(`if`(irem(d-4, 2, 'r')=1, d*r, 0)
%p A263395        , d=divisors(j))*a(n-j), j=1..n)/n)
%p A263395     end:
%p A263395 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 17 2015
%t A263395 nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+5))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263395 nmax = 60; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263395 Cf. A035528, A263150, A263352, A263396, A263397.
%K A263395 nonn
%O A263395 0,10
%A A263395 _Vaclav Kotesovec_, Oct 16 2015