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A365525 a(n) = Sum_{k=0..floor(n/4)} Stirling2(n,4*k).

%I A365525 #27 Jun 10 2025 07:32:47
%S A365525 1,0,0,0,1,10,65,350,1702,7806,34855,157630,770529,4432220,31307432,
%T A365525 259090260,2316320073,21172354778,193091210857,1744478148866,
%U A365525 15627203762926,139526376391986,1251976261264071,11417796498945894,107280845105151601
%N A365525 a(n) = Sum_{k=0..floor(n/4)} Stirling2(n,4*k).
%H A365525 Robert Israel, <a href="/A365525/b365525.txt">Table of n, a(n) for n = 0..574</a>
%F A365525 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 A365525 G.f.: Sum_{k>=0} x^(4*k) / Product_{j=1..4*k} (1-j*x).
%F A365525 a(n) ~ n^n / (4 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - _Vaclav Kotesovec_, Jun 10 2025
%p A365525 f:= proc(n) local k; add(Stirling2(n,4*k),k=0..n/4) end proc:
%p A365525 map(f, [$0..30]); # _Robert Israel_, Sep 11 2024
%t A365525 a[n_] := Sum[StirlingS2[n, 4*k], {k, 0, Floor[n/4]}]; Array[a, 25, 0] (* _Amiram Eldar_, Sep 11 2023 *)
%o A365525 (PARI) a(n) = sum(k=0, n\4, stirling(n, 4*k, 2));
%o A365525 (Python)
%o A365525 from sympy.functions.combinatorial.numbers import stirling
%o A365525 def A365525(n): return sum(stirling(n,k<<2) for k in range((n>>2)+1)) # _Chai Wah Wu_, Sep 08 2023
%Y A365525 Cf. A024430, A143815, A365528.
%Y A365525 Cf. A099948, A365526, A365527.
%K A365525 nonn
%O A365525 0,6
%A A365525 _Seiichi Manyama_, Sep 08 2023