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A365530 a(n) = Sum_{k=0..floor((n-2)/5)} Stirling2(n,5*k+2).

%I A365530 #8 Sep 08 2023 07:22:57
%S A365530 0,0,1,3,7,15,31,64,155,717,6391,65010,629444,5719597,49340838,
%T A365530 408864186,3284672489,25770192646,198718943490,1516391860879,
%U A365530 11554571944615,89144035246500,711587142257776,6054854693784594,56609279400922224,590143167134961765
%N A365530 a(n) = Sum_{k=0..floor((n-2)/5)} Stirling2(n,5*k+2).
%F A365530 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), A365529(n) = B(n), a(n) = C(n), A365531(n) = D(n) and A365532(n) = E(n).
%F A365530 G.f.: Sum_{k>=0} x^(5*k+2) / Product_{j=1..5*k+2} (1-j*x).
%o A365530 (PARI) a(n) = sum(k=0, (n-2)\5, stirling(n, 5*k+2, 2));
%Y A365530 Cf. A365528, A365529, A365531, A365532.
%K A365530 nonn
%O A365530 0,4
%A A365530 _Seiichi Manyama_, Sep 08 2023