A352118 Expansion of e.g.f. 1/(2 - exp(3*x))^(1/3).

1, 1, 7, 73, 1063, 20041, 464167, 12752713, 405439783, 14641740361, 592050220327, 26499885031753, 1300723181304103, 69470729022993481, 4010891467932629287, 248920020505516389193, 16525139232054244298023, 1168557027163488299171401

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

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

a(n) ~ n! * 3^n / (2^(1/3) * Gamma(1/3) * n^(2/3) * log(2)^(n + 1/3)). - Vaclav Kotesovec, Mar 05 2022

From Seiichi Manyama, Nov 18 2023: (Start)

a(0) = 1; a(n) = Sum_{k=1..n} 3^k * (1 - 2/3 * k/n) * binomial(n,k) * a(n-k).

a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-3)^k * binomial(n-1,k) * a(n-k). (End)

Mathematica

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

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(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));

Crossrefs

Cf. A000670, A352117, A352119.

Cf. A352070.

Sequence in context: A112939 A058350 A048174 * A258379 A325930 A360544

Adjacent sequences: A352115 A352116 A352117 * A352119 A352120 A352121

Keyword

nonn

Author

Seiichi Manyama, Mar 05 2022

Status

approved