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

%I A365528 #18 Jun 10 2025 07:33:45
%S A365528 1,0,0,0,0,1,15,140,1050,6951,42526,246785,1381105,7547826,40827787,
%T A365528 223429571,1289945660,8411093621,66070626548,624900235273,
%U A365528 6667243384356,74991482322466,854627237256694,9698297591786441,108934902927646609
%N A365528 a(n) = Sum_{k=0..floor(n/5)} Stirling2(n,5*k).
%F A365528 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).
%F A365528 G.f.: Sum_{k>=0} x^(5*k) / Product_{j=1..5*k} (1-j*x).
%F A365528 a(n) ~ n^n / (5 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - _Vaclav Kotesovec_, Jun 10 2025
%t A365528 a[n_] := Sum[StirlingS2[n, 5*k], {k, 0, Floor[n/5]}]; Array[a, 25, 0] (* _Amiram Eldar_, Sep 13 2023 *)
%o A365528 (PARI) a(n) = sum(k=0, n\5, stirling(n, 5*k, 2));
%Y A365528 Cf. A365529, A365530, A365531, A365532.
%Y A365528 Cf. A024430, A143815, A365525.
%K A365528 nonn
%O A365528 0,7
%A A365528 _Seiichi Manyama_, Sep 08 2023