A365557 - OEIS

%I #24 Nov 17 2023 11:20:29

%S 1,5,60,1105,27505,862900,32665935,1448431605,73618245530,

%T 4219213176975,269178309769385,18919087590749230,1452439246800583805,

%U 120926788470961893425,10852505784073190637460,1044349665968997385498605,107273533723839304683589205

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

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

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

%F 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

%F 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

%F 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

%t 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 *)

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

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

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 09 2023