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A365529 a(n) = Sum_{k=0..floor((n-1)/5)} Stirling2(n,5*k+1).

%I A365529 #8 Sep 08 2023 07:23:02
%S A365529 0,1,1,1,1,1,2,22,267,2647,22828,179489,1323719,9323744,63502440,
%T A365529 422172752,2763863468,18017811013,119078265944,822495346707,
%U A365529 6206943675825,53413341096271,529613886789747,5863983528090106,69211078916780252,839908976768680556
%N A365529 a(n) = Sum_{k=0..floor((n-1)/5)} Stirling2(n,5*k+1).
%F A365529 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). A365528(n) = A(n), a(n) = B(n), A365530(n) = C(n), A365531(n) = D(n) and A365532(n) = E(n).
%F A365529 G.f.: Sum_{k>=0} x^(5*k+1) / Product_{j=1..5*k+1} (1-j*x).
%o A365529 (PARI) a(n) = sum(k=0, (n-1)\5, stirling(n, 5*k+1, 2));
%Y A365529 Cf. A365528, A365530, A365531, A365532.
%K A365529 nonn
%O A365529 0,7
%A A365529 _Seiichi Manyama_, Sep 08 2023