A361008 G.f.: Product_{k >= 0} ((1 + x^(2*k+1)) / (1 - x^(2*k+1)))^k.

1, 0, 0, 2, 0, 4, 2, 6, 8, 10, 20, 18, 42, 40, 78, 92, 140, 192, 258, 382, 480, 728, 902, 1334, 1698, 2404, 3148, 4292, 5742, 7608, 10304, 13430, 18192, 23592, 31720, 41144, 54766, 71188, 93762, 122156, 159420, 207820, 269380, 350726, 452434, 587520, 755446

Offset: 0

Author

Vaclav Kotesovec, Apr 20 2023

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A080054, A291697, A263149, A263150.

Cf. A156616, A270919.

Programs

Mathematica

Table[SeriesCoefficient[Product[((1 + x^(2*k + 1))/(1 - x^(2*k + 1)))^k, {k, 0, n}], {x, 0, n}], {n, 0, 50}]

Formula

a(n) ~ sqrt(A/(3*Pi)) * (7*zeta(3))^(11/72) * exp(3*(7*zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3)/(8*(7*zeta(3))^(1/3)) - 1/24 - Pi^4/(1344*zeta(3))) / (2^(3/4) * n^(47/72)), where A = A074962 is the Glaisher-Kinkelin constant.