A351737 Expansion of e.g.f. exp( x * (exp(3 * x) - 1) ).

1, 0, 6, 27, 216, 2025, 21708, 260253, 3460320, 50395041, 795324420, 13495904829, 244747554912, 4718754452529, 96285948702804, 2071265238290565, 46815054101658432, 1108489016781839169, 27424412680091114628, 707277138662880504045, 18974871706141125008640

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..461

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * Stirling2(n-k,k)/(n-k)!.

From Seiichi Manyama, Aug 29 2022: (Start)

a(n) = Sum_{k=0..n} (3*k-1)^(n-k) * binomial(n,k).

G.f.: Sum_{k>=0} x^k / (1 - (3*k-1)*x)^(k+1). (End)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(3*x)-1))))
(PARI) a(n) = n!*sum(k=0, n\2, 3^(n-k)*stirling(n-k, k, 2)/(n-k)!);
(PARI) a(n) = sum(k=0, n, (3*k-1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022
(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(3*k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

Crossrefs

Cf. A052506, A351736.

Cf. A351734, A351735.

Sequence in context: A289022 A060977 A267630 * A047778 A290802 A360754

Adjacent sequences: A351734 A351735 A351736 * A351738 A351739 A351740

Keyword

nonn

Author

Seiichi Manyama, May 20 2022

Status

approved