A365528 a(n) = Sum_{k=0..floor(n/5)} Stirling2(n,5*k).

1, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42526, 246785, 1381105, 7547826, 40827787, 223429571, 1289945660, 8411093621, 66070626548, 624900235273, 6667243384356, 74991482322466, 854627237256694, 9698297591786441, 108934902927646609

Offset: 0

Author

Seiichi Manyama, Sep 08 2023

Keywords

• nonn

Crossrefs

Cf. A365529, A365530, A365531, A365532.

Cf. A024430, A143815, A365525.

Programs

Mathematica

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));

Formula

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