A365599 Expansion of e.g.f. 1 / (1 - 3 * log(1 + x))^(2/3).

1, 2, 8, 54, 498, 5868, 83940, 1413480, 27375240, 599437440, 14641665120, 394657325280, 11635613604000, 372469741813440, 12864889063033920, 476870475257550720, 18882021780125953920, 795381867831610978560, 35515223076159203880960

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j+2)) * Stirling1(n,k).

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

a(n) ~ Gamma(1/3) * n^(n + 1/6) / (3^(1/6) * sqrt(2*Pi) * (exp(1/3) - 1)^(n + 2/3) * exp(n - 2/9)). - Vaclav Kotesovec, Nov 11 2023

Mathematica

a[n_] := Sum[Product[3*j + 2, {j, 0, k - 1}] * StirlingS1[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Sep 13 2023 *)

Prog

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

Crossrefs

Cf. A335531, A347020.

Cf. A365575.

Sequence in context: A352648 A052662 A375224 * A199576 A005155 A133316

Adjacent sequences: A365596 A365597 A365598 * A365600 A365601 A365602

Keyword

nonn

Author

Seiichi Manyama, Sep 11 2023

Status

approved