A306081 Expansion of e.g.f. Product_{k>=1} ((1 + (exp(x) - 1)^k) / (1 - (exp(x) - 1)^k))^k.

1, 2, 14, 134, 1574, 22262, 367814, 6907574, 144942854, 3357588662, 85000841414, 2331998188214, 68862337593734, 2176283210561462, 73250933670041414, 2614843434740912054, 98632371931151518214, 3918608865052986708662, 163507638190268814991814

Offset: 0

Author

Vaclav Kotesovec, Jun 20 2018

Keywords

• nonn

Comments

Convolution of A306080 and A306046.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A156616, A305550, A306045, A306046, A306080.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Product[((1 + (Exp[x] - 1)^k)/(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) * A156616(k) * k!.

a(n) ~ n! * exp(3 * (7*Zeta(3))^(1/3) * n^(2/3) / (4 * log(2)^(2/3)) + (1 - log(2)) * (7*Zeta(3))^(2/3) * n^(1/3) / (8 * log(2)^(4/3)) - 7*(log(2)^2 + log(2) - 1) * Zeta(3) / (48 * log(2)^2) + 1/12) * (7*Zeta(3))^(7/36) / (A * 2^(13/12) * sqrt(3*Pi) * n^(25/36) * (log(2))^(n + 11/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jun 22 2018