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%I #15 Sep 11 2023 01:46:18
%S 1,0,0,0,1,10,65,350,1702,7806,34855,157630,770529,4432220,31307432,
%T 259090260,2316320073,21172354778,193091210857,1744478148866,
%U 15627203762926,139526376391986,1251976261264071,11417796498945894,107280845105151601
%N a(n) = Sum_{k=0..floor(n/4)} Stirling2(n,4*k).
%F Let A(0)=1, B(0)=0, C(0)=0 and D(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) and A(n+1) = Sum_{k=0..n} binomial(n,k)*D(k). a(n) = A(n), A365526(n) = B(n), A365527(n) = C(n) and A099948(n) = D(n).
%F G.f.: Sum_{k>=0} x^(4*k) / Product_{j=1..4*k} (1-j*x).
%t a[n_] := Sum[StirlingS2[n, 4*k], {k, 0, Floor[n/4]}]; Array[a, 25, 0] (* _Amiram Eldar_, Sep 11 2023 *)
%o (PARI) a(n) = sum(k=0, n\4, stirling(n, 4*k, 2));
%o (Python)
%o from sympy.functions.combinatorial.numbers import stirling
%o def A365525(n): return sum(stirling(n,k<<2) for k in range((n>>2)+1)) # _Chai Wah Wu_, Sep 08 2023
%Y Cf. A099948, A365526, A365527.
%K nonn
%O 0,6
%A _Seiichi Manyama_, Sep 08 2023
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