A263396 Expansion of Product_{k>=1} 1/(1 - x^(2*k+7))^k.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 1, 6, 2, 7, 6, 8, 10, 9, 19, 11, 28, 13, 44, 18, 60, 27, 85, 42, 111, 67, 148, 109, 188, 169, 245, 260, 313, 390, 408, 568, 535, 811, 717, 1139, 974, 1568, 1343, 2134, 1872, 2873, 2621, 3832, 3687, 5088

Offset: 0

Author

Vaclav Kotesovec, Oct 16 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Crossrefs

Cf. A035528, A263150, A263352, A263395, A263397.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-6, 2, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..65);  # Alois P. Heinz, Oct 17 2015

Mathematica

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+7))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(9*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(9*k)/(k*(1-x^(2*k))^2)).

a(n) ~ 8 * 2^(1/72) * exp(-1/24 - 49*Pi^4/(1728*Zeta(3)) - 7*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^(97/72) / (45*sqrt(3*Pi) * Zeta(3)^(133/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.