A306083 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^(k^2)).

1, 1, 1, 1, 25, 361, 3361, 25201, 166825, 1383481, 25879921, 651816001, 14450460025, 280347467401, 5253918022081, 107822784560401, 2578135250199625, 69030779356572121, 1953531819704493841, 56903093167217522401, 1689294590583626265625

Offset: 0

Author

Vaclav Kotesovec, Jun 20 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Crossrefs

Cf. A033461, A306082, A306147.

Programs

Maple

a:=series(mul(1+(exp(x)-1)^(k^2),k=1..100),x=0,21): seq(n!*coeff(a, x, n),n=0..20); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A033461(k) * k!.

a(n) ~ n! * exp(3 * (Pi/log(2))^(1/3) * ((sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 4) * ((sqrt(2) - 1) * Zeta(3/2) / Pi)^(1/3) / (2 * sqrt(6) * n^(5/6) * log(2)^(n + 1/6)).