A263363 Expansion of Product_{k>=1} 1/(1-x^(k+7))^k.

1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 17, 22, 32, 42, 59, 76, 103, 130, 171, 216, 280, 354, 460, 584, 757, 968, 1249, 1596, 2056, 2618, 3354, 4266, 5441, 6900, 8778, 11108, 14094, 17814, 22546, 28450, 35946, 45280, 57088, 71806, 90347

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. A000219, A052847, A263358, A263359, A263360, A263361, A263362, A263364.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-7), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 16 2015

Mathematica

nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+7))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(8*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(8*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - 49*Pi^4/(432*Zeta(3)) - 7*Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(269/36) * Pi^3 / (398131200 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(287/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.