A306046 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^k.

1, 1, 7, 55, 571, 6991, 101467, 1682815, 31370731, 648823951, 14728727227, 363609116575, 9692252794891, 277304683729711, 8471938268282587, 275137855204310335, 9461893931226763051, 343394421233354232271, 13112532730352768439547, 525396814643685317840095

Offset: 0

Author

Vaclav Kotesovec, Jun 18 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400
• Vaclav Kotesovec, Graph - The asymptotic ratio (20000 terms)

Crossrefs

Cf. A000219, A167137, A305550, A306045, A306081.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A000219(k) * k!.

a(n) ~ n! * exp(3 * Zeta(3)^(1/3) * n^(2/3) / (2^(4/3) * log(2)^(2/3)) + (1 - log(2)) * Zeta(3)^(2/3) * n^(1/3) / (2^(5/3) * log(2)^(4/3)) - (log(2)^2 + log(2) - 1) * Zeta(3) / (12 * log(2)^2) + 1/12) * Zeta(3)^(7/36) / (A * 2^(11/18) * sqrt(3*Pi) * n^(25/36) * (log(2))^(n + 11/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jun 22 2018