A365556 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(2/3).

1, 4, 44, 764, 18204, 551644, 20291804, 877970524, 43680345564, 2456429581404, 154072160204764, 10663000409493084, 807124301044917724, 66329628496719183964, 5881222650127663682524, 559616682597652939940444, 56879286407092006924382684

Offset

0,2

Links

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

Formula

a(n) = Sum_{k=0..n} 2^k * (Product_{j=0..k-1} (3*j+2)) * Stirling2(n,k) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+4)) * Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (6 - 2*k/n) * binomial(n,k) * a(n-k).

O.g.f. (conjectural): 1/(1 - 4*x/(1 - 7*x/(1 - 10*x/(1 - 14*x/(1 - 16*x/(1 - 21*x/(1 - ... - (6*n - 2)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Sep 24 2023

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

a(0) = 1; a(n) = 4*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 + 4, {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+4)*stirling(n, k, 2));

Crossrefs

Cf. A094419, A346985, A354252, A365555, A365557.

Sequence in context: A053332 A276369 A157093 * A088594 A144827 A354260

Adjacent sequences: A365553 A365554 A365555 * A365557 A365558 A365559

Keyword

nonn

Author

Seiichi Manyama, Sep 09 2023

Status

approved