A262811 Expansion of Product_{k>=1} 1/(1-x^(2*k-1))^(2*k-1).

1, 1, 1, 4, 4, 9, 15, 22, 37, 56, 92, 133, 210, 310, 466, 696, 1013, 1495, 2160, 3141, 4495, 6462, 9172, 13024, 18387, 25840, 36213, 50500, 70280, 97302, 134522, 185105, 254245, 347938, 475036, 646676, 878145, 1189468, 1607095, 2166672, 2913794, 3910741

Offset: 0

Author

Vaclav Kotesovec, Oct 03 2015

Keywords

• nonn

Links

• Alois P. Heinz, 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, A003293, A035528, A161870, A262736, A292038.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d::even, 0, d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Oct 05 2015

Mathematica

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

Formula

a(n) ~ exp(-1/12 + 3*Zeta(3)^(1/3)*n^(2/3)/2) * A * Zeta(3)^(5/36) / (2^(2/3) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A050999(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 09 2017