A263199 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^(2*k+1).

1, 0, 0, 3, 0, 5, 6, 7, 15, 19, 36, 41, 77, 100, 156, 230, 317, 482, 665, 981, 1354, 1967, 2710, 3852, 5363, 7453, 10373, 14287, 19780, 27022, 37220, 50583, 69140, 93693, 127098, 171640, 231469, 311323, 417627, 559577, 747122, 996947, 1325872, 1761900

Offset: 0

Author

Vaclav Kotesovec, Oct 12 2015

Keywords

• nonn

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. A000219, A035528, A262811, A263140, A263149, A263150.

Programs

Maple

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

Mathematica

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

Formula

For n>1, a(n) = A262811(n) - A262811(n-1).

a(n) ~ A * Zeta(3)^(17/36) * exp(-1/12 + 3 * Zeta(3)^(1/3) * n^(2/3)/2) / (2^(2/3) * sqrt(3*Pi) * n^(35/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.