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A365531 a(n) = Sum_{k=0..floor((n-3)/5)} Stirling2(n,5*k+3).

%I A365531 #8 Sep 08 2023 07:22:52
%S A365531 0,0,0,1,6,25,90,301,967,3061,10080,40381,245553,2161238,21701381,
%T A365531 219007491,2149071359,20442363031,189226358659,1712836890912,
%U A365531 15232581945180,133717667932475,1164901223314180,10143255631462661,89207257764369032,804712211338739040
%N A365531 a(n) = Sum_{k=0..floor((n-3)/5)} Stirling2(n,5*k+3).
%F A365531 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), A365530(n) = C(n), a(n) = D(n) and A365532(n) = E(n).
%F A365531 G.f.: Sum_{k>=0} x^(5*k+3) / Product_{j=1..5*k+3} (1-j*x).
%o A365531 (PARI) a(n) = sum(k=0, (n-3)\5, stirling(n, 5*k+3, 2));
%Y A365531 Cf. A365528, A365529, A365530, A365532.
%K A365531 nonn
%O A365531 0,5
%A A365531 _Seiichi Manyama_, Sep 08 2023