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A263362 Expansion of Product_{k>=1} 1/(1-x^(k+6))^k.

%I A263362 #14 Nov 16 2024 16:10:21
%S A263362 1,0,0,0,0,0,0,1,2,3,4,5,6,7,9,11,16,21,31,41,58,76,103,133,178,229,
%T A263362 303,394,519,675,889,1155,1513,1964,2558,3310,4298,5543,7169,9231,
%U A263362 11903,15289,19665,25208,32339,41374,52943,67595,86307,109965,140089,178155
%N A263362 Expansion of Product_{k>=1} 1/(1-x^(k+6))^k.
%H A263362 Vaclav Kotesovec, <a href="/A263362/b263362.txt">Table of n, a(n) for n = 0..5000</a>
%H A263362 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 A263362 G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^k)^2)).
%F A263362 a(n) ~ exp(1/12 - Pi^4/(12*Zeta(3)) - Pi^2 * n^(1/3) / (2^(1/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(191/36) * Pi^(5/2) / (276480 * A * 2^(11/36) * sqrt(3) * Zeta(3)^(209/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263362 with(numtheory):
%p A263362 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263362       max(0, d-6), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263362     end:
%p A263362 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 16 2015
%t A263362 nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+6))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263362 nmax = 60; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263362 Cf. A000219, A052847, A263358, A263359, A263360, A263361, A263363, A263364.
%K A263362 nonn
%O A263362 0,9
%A A263362 _Vaclav Kotesovec_, Oct 16 2015