A263395 Expansion of Product_{k>=1} 1/(1 - x^(2*k+5))^k.

1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 1, 5, 2, 6, 6, 7, 10, 9, 19, 11, 28, 16, 44, 25, 61, 40, 87, 65, 116, 107, 160, 168, 215, 260, 295, 393, 407, 578, 573, 836, 814, 1193, 1167, 1675, 1684, 2335, 2427, 3238, 3501, 4468, 5014, 6161, 7152, 8494, 10121

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, A263396, A263397.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-4, 2, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015

Mathematica

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+5))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^(2*k))^2)).

a(n) ~ 2^(109/72) * exp(-1/24 - 25*Pi^4/(1728*Zeta(3)) - 5*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^(25/72) / (3*sqrt(3*Pi) * Zeta(3)^(61/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.