A365570 - OEIS

%I #18 Nov 16 2023 11:51:27

%S 1,4,40,616,12856,338728,10781176,402250216,17213590840,831013114792,

%T 44675458306168,2646758624166760,171319908334752184,

%U 12028779733435667752,910538645035885918456,73918475291961325824232,6406179168820339231897144

%N Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(4/5).

%F a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+4)) * Stirling2(n,k).

%F a(0) = 1; a(n) = Sum_{k=1..n} (5 - k/n) * binomial(n,k) * a(n-k).

%F a(n) ~ sqrt(2*Pi) * n^(n + 3/10) / (6^(4/5) * Gamma(4/5) * exp(n) * log(6/5)^(n + 4/5)). - _Vaclav Kotesovec_, Nov 11 2023

%F a(0) = 1; a(n) = 4*a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - _Seiichi Manyama_, Nov 16 2023

%t a[n_] := Sum[Product[5*j + 4, {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, 5*j+4)*stirling(n, k, 2));

%Y Cf. A094418, A346984, A365568, A365569.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 09 2023