A374882 Expansion of e.g.f. exp( (1 - (1 - 9*x)^(1/3))/3 ).

1, 1, 7, 109, 2665, 88981, 3768391, 193406977, 11663021329, 808092594505, 63252127883431, 5519514702282901, 531266903931402937, 55912682968563924829, 6387276499619184590695, 787104141893585220839401, 104074098535487279656795681, 14697203663694095986066104337

Offset

0,3

Links

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

Eric Weisstein's World of Mathematics, Bell Polynomial.

Formula

a(n) = Sum_{k=0..n} (-9)^(n-k) * Stirling1(n,k) * A317996(k) = (-9)^n * Sum_{k=0..n} (1/3)^k * Stirling1(n,k) * Bell_k(-1/3), where Bell_n(x) is n-th Bell polynomial.

From Vaclav Kotesovec, Aug 02 2024: (Start)

a(n) = 18*(n-2)*a(n-1) - 9*(3*n-8)*(3*n-7)*a(n-2) + a(n-3).

a(n) ~ Gamma(1/3) * 3^(2*n - 3/2) * n^(n - 5/6) / (sqrt(2*Pi) * exp(n - 1/3)) * (1 - 2*Pi/(3^(3/2)*Gamma(1/3)^2*n^(1/3))). (End)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1-(1-9*x)^(1/3))/3)))

Crossrefs

Cf. A080893, A252284, A317996.

Cf. A375174, A375176.

Sequence in context: A239848 A274787 A116875 * A357395 A371317 A303109

Adjacent sequences: A374879 A374880 A374881 * A374883 A374884 A374885

Keyword

nonn

Author

Seiichi Manyama, Aug 02 2024

Status

approved