A263362 Expansion of Product_{k>=1} 1/(1-x^(k+6))^k.

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, 303, 394, 519, 675, 889, 1155, 1513, 1964, 2558, 3310, 4298, 5543, 7169, 9231, 11903, 15289, 19665, 25208, 32339, 41374, 52943, 67595, 86307, 109965, 140089, 178155

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, A263363, A263364.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-6), 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+6))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

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.