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A365526 a(n) = Sum_{k=0..floor((n-1)/4)} Stirling2(n,4*k+1).

%I A365526 #16 Sep 11 2024 14:28:41
%S A365526 0,1,1,1,1,2,16,141,1051,6953,42571,247886,1401676,7868005,45210257,
%T A365526 277899961,1917140421,15186484134,135259346092,1295096363273,
%U A365526 12821558136891,128268683204737,1283599391456735,12817818177339530,127998022119881272
%N A365526 a(n) = Sum_{k=0..floor((n-1)/4)} Stirling2(n,4*k+1).
%H A365526 Robert Israel, <a href="/A365526/b365526.txt">Table of n, a(n) for n = 0..574</a>
%F A365526 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). A365525(n) = A(n), a(n) = B(n), A365527(n) = C(n) and A099948(n) = D(n).
%F A365526 G.f.: Sum_{k>=0} x^(4*k+1) / Product_{j=1..4*k+1} (1-j*x).
%p A365526 f:= proc(n) local k; add(Stirling2(n,4*k+1),k=0..(n-1)/4) end proc:
%p A365526 map(f, [$0..30]); # _Robert Israel_, Sep 11 2024
%t A365526 a[n_] := Sum[StirlingS2[n, 4*k+1], {k, 0, Floor[(n-1)/4]}]; Array[a, 25, 0] (* _Amiram Eldar_, Sep 13 2023 *)
%o A365526 (PARI) a(n) = sum(k=0, (n-1)\4, stirling(n, 4*k+1, 2));
%Y A365526 Cf. A099948, A365525, A365527.
%K A365526 nonn
%O A365526 0,6
%A A365526 _Seiichi Manyama_, Sep 08 2023