0,7

Let A(0)=1, B(0)=0, C(0)=0, D(0)=0 and E(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), D(n+1) = Sum_{k=0..n} binomial(n,k)*C(k), E(n+1) = Sum_{k=0..n} binomial(n,k)*D(k) and A(n+1) = Sum_{k=0..n} binomial(n,k)*E(k). a(n) = A(n), A365529(n) = B(n), A365530(n) = C(n), A365531(n) = D(n) and A365532(n) = E(n).

G.f.: Sum_{k>=0} x^(5*k) / Product_{j=1..5*k} (1-j*x).

a(n) ~ n^n / (5 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Jun 10 2025

a[n_] := Sum[StirlingS2[n, 5*k], {k, 0, Floor[n/5]}]; Array[a, 25, 0] (* Amiram Eldar, Sep 13 2023 *)

(PARI) a(n) = sum(k=0, n\5, stirling(n, 5*k, 2));

Cf. A365529, A365530, A365531, A365532.

Cf. A024430, A143815, A365525.

Sequence in context: A354398 A056281 A000481 * A327506 A346955 A346920

Adjacent sequences: A365525 A365526 A365527 * A365529 A365530 A365531

Seiichi Manyama, Sep 08 2023

approved