This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365527 #13 Sep 13 2023 02:09:54
%S A365527 0,0,1,3,7,15,32,84,393,2901,23339,180565,1327404,9364732,64197317,
%T A365527 433372411,2928720335,20264399483,147807954692,1170622475408,
%U A365527 10229966924581,97922117830589,1001744359476291,10661002700183905,115706501336004984
%N A365527 a(n) = Sum_{k=0..floor((n-2)/4)} Stirling2(n,4*k+2).
%F A365527 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), A365526(n) = B(n), a(n) = C(n) and A099948(n) = D(n).
%F A365527 G.f.: Sum_{k>=0} x^(4*k+2) / Product_{j=1..4*k+2} (1-j*x).
%t A365527 a[n_] := Sum[StirlingS2[n, 4*k+2], {k, 0, Floor[(n-2)/4]}]; Array[a, 25, 0] (* _Amiram Eldar_, Sep 13 2023 *)
%o A365527 (PARI) a(n) = sum(k=0, (n-2)\4, stirling(n, 4*k+2, 2));
%Y A365527 Cf. A099948, A365525, A365526.
%K A365527 nonn
%O A365527 0,4
%A A365527 _Seiichi Manyama_, Sep 08 2023