A365557 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(5/6).

1, 5, 60, 1105, 27505, 862900, 32665935, 1448431605, 73618245530, 4219213176975, 269178309769385, 18919087590749230, 1452439246800583805, 120926788470961893425, 10852505784073190637460, 1044349665968997385498605, 107273533723839304683589205

Offset

0,2

Links

Table of n, a(n) for n=0..16.

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+5)) * Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (6 - k/n) * binomial(n,k) * a(n-k).

O.g.f. (conjectural): 1/(1 - 5*x/(1 - 7*x/(1 - 11*x/(1 - 14*x/(1 - 17*x/(1 - 21*x/(1 - ... - (6*n - 1)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Sep 24 2023

a(n) ~ Gamma(1/3)^2 * sqrt(3) * n^(n + 1/3) / (14^(5/6) * Pi * exp(n) * log(7/6)^(n + 5/6)). - Vaclav Kotesovec, Nov 11 2023

a(0) = 1; a(n) = 5*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

Mathematica

a[n_] := Sum[Product[6*j + 5, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)

Prog

(PARI) a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+5)*stirling(n, k, 2));

Crossrefs

Cf. A094419, A346985, A354252, A365555, A365556.

Sequence in context: A128574 A120976 A062980 * A113665 A147585 A138215

Adjacent sequences: A365554 A365555 A365556 * A365558 A365559 A365560

Keyword

nonn

Author

Seiichi Manyama, Sep 09 2023

Status

approved