A306080 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k)^k.

1, 1, 5, 43, 401, 4651, 64265, 1015603, 17996081, 354373531, 7682286425, 181466541763, 4632985312961, 127068851847211, 3724903637434985, 116185013450349523, 3840969677266089041, 134113334651486325691, 4930511086446971405945, 190327859758408148070883

Offset: 0

Author

Vaclav Kotesovec, Jun 20 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..410
• Vaclav Kotesovec, Graph - The asymptotic ratio

Crossrefs

Cf. A026007, A305550, A306046, A306081.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Product[(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) * A026007(k) * k!.

a(n) ~ n! * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (4 * log(2)^(2/3)) + (1 - log(2)) * (3*Zeta(3))^(2/3) * n^(1/3) / (8 * log(2)^(4/3)) - (log(2)^2 + log(2) - 1) * Zeta(3) / (16 * log(2)^2)) * Zeta(3)^(1/6) / (2^(13/12) * 3^(1/3) * sqrt(Pi) * n^(2/3) * (log(2))^(n + 1/3)). - Vaclav Kotesovec, Jun 23 2018