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Expansion of Product_{k>=1} 1/(1-x^(k+3))^k.
8

%I #11 Oct 17 2015 04:55:25

%S 1,0,0,0,1,2,3,4,6,8,13,18,29,40,61,86,127,178,260,364,524,734,1042,

%T 1454,2051,2848,3981,5510,7652,10542,14558,19970,27428,37480,51222,

%U 69720,94870,128634,174306,235506,317899,428018,575688,772540,1035538,1385264

%N Expansion of Product_{k>=1} 1/(1-x^(k+3))^k.

%H Vaclav Kotesovec, <a href="/A263359/b263359.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^(4*k)/(k*(1-x^k)^2).

%F a(n) ~ exp(1/12 - Pi^4/(48*Zeta(3)) - Pi^2 * n^(1/3) / (2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(29/36) * Pi / (A * 2^(47/36) * sqrt(3) * Zeta(3)^(47/36)), 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-3), d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Oct 16 2015

%t nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+3))^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 50; CoefficientList[Series[E^Sum[x^(4*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000219, A052847, A263358, A263360, A263361, A263362, A263363, A263364.

%K nonn

%O 0,6

%A _Vaclav Kotesovec_, Oct 16 2015