cp's OEIS Frontend

A263360 Expansion of Product_{k>=1} 1/(1-x^(k+4))^k.

%I A263360 #14 Nov 16 2024 16:10:40
%S A263360 1,0,0,0,0,1,2,3,4,5,7,9,14,19,29,40,58,79,113,153,215,294,407,555,
%T A263360 767,1040,1424,1930,2624,3540,4794,6441,8677,11627,15589,20818,27812,
%U A263360 37011,49257,65360,86681,114665,151594,199947,263530,346647,455553,597628
%N A263360 Expansion of Product_{k>=1} 1/(1-x^(k+4))^k.
%H A263360 Vaclav Kotesovec, <a href="/A263360/b263360.txt">Table of n, a(n) for n = 0..5000</a>
%H A263360 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 A263360 G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1-x^k)^2)).
%F A263360 a(n) ~ exp(1/12 - Pi^4/(27*Zeta(3)) - 2^(2/3) * Pi^2 * n^(1/3) / (3 * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(71/36) * Pi^(3/2) / (12 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(89/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263360 with(numtheory):
%p A263360 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263360       max(0, d-4), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263360     end:
%p A263360 seq(a(n), n=0..50);  # _Alois P. Heinz_, Oct 16 2015
%t A263360 nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+4))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263360 nmax = 50; CoefficientList[Series[E^Sum[x^(5*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263360 Cf. A000219, A052847, A263358, A263359, A263361, A263362, A263363, A263364.
%K A263360 nonn
%O A263360 0,7
%A A263360 _Vaclav Kotesovec_, Oct 16 2015