A263150 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^k.

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 6, 7, 11, 12, 21, 22, 34, 38, 59, 67, 95, 118, 155, 198, 252, 330, 409, 540, 662, 867, 1067, 1382, 1705, 2187, 2705, 3430, 4267, 5348, 6666, 8303, 10352, 12812, 15964, 19681, 24467, 30091, 37282, 45769, 56539, 69296, 85304

Offset: 0

Author

Vaclav Kotesovec, Oct 10 2015

Keywords

• nonn

Comments

The side effect of this calculation is a formula: Integral_{x=0..infinity} exp(-3*x)/(x*(1-exp(-2*x))^2) - 1/(4*x^3) + 1/(4*x^2) - exp(-x)/(24*x) = log(2)/6 + log(A)/2 - 1/24, where A = A074962 is the Glaisher-Kinkelin constant.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• 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, A052847, A263140, A263149, A263199, A263352, A263395, A263396, A263397.

Programs

Maple

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

Mathematica

nmax = 100; CoefficientList[Series[Product[1/(1-x^(2*k+1))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(2*j))^2).

a(n) ~ sqrt(A) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(8/3)* Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)/2) / (2^(35/72) * sqrt(3*Pi) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.