%I #11 Oct 17 2015 04:54:37

%S 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 303,394,519,675,889,1155,1513,1964,2558,3310,4298,5543,7169,9231,

%U 11903,15289,19665,25208,32339,41374,52943,67595,86307,109965,140089,178155

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

%H Vaclav Kotesovec, <a href="/A263362/b263362.txt">Table of n, a(n) for n = 0..5000</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_{k>=1} x^(7*k)/(k*(1-x^k)^2).

%F 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 with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

%p max(0, d-6), d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Oct 16 2015

%t nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+6))^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 60; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000219, A052847, A263358, A263359, A263360, A263361, A263363, A263364.

%K nonn

%O 0,9

%A _Vaclav Kotesovec_, Oct 16 2015