A352122 Expansion of e.g.f. (2 - exp(-3*x))^(1/3).

1, 1, -5, 37, -413, 6421, -128285, 3125557, -89781053, 2969440021, -111109062365, 4639580153077, -213856576973693, 10784605095793621, -590598038062108445, 34901993971832092597, -2213771863243583654333, 150004882482828402563221

Offset: 0

Author

Seiichi Manyama, Mar 05 2022

Keywords

• sign

Crossrefs

Cf. A352121, A352123.

Cf. A352113.

Programs

Mathematica

m = 17; Range[0, m]! * CoefficientList[Series[(2 - Exp[-3*x])^(1/3), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace((2-exp(-3*x))^(1/3)))

PARI

a(n) = sum(k=0, n, (-3)^(n-k)*prod(j=0, k-1, -3*j+1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-3)^(n-k) * (Product_{j=0..k-1} (-3*j+1)) * Stirling2(n,k).

a(n) ~ n! * (-1)^(n+1) * Gamma(1/3) * 3^(n - 1/2) / (Pi * 2^(2/3) * n^(4/3) * log(2)^(n - 1/3)). - Vaclav Kotesovec, Mar 06 2022