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 A344397 #16 Dec 23 2024 21:50:50
%S A344397 1,0,1,1,14,30,540,1806,40824,186480,5103000,29607600,953029440,
%T A344397 6711344640,248619571200,2060056318320,86355926616960,823172919528960,
%U A344397 38528927611574400,415357755774998400,21473732319740064000,258323865658578720000,14620825330739032204800
%N A344397 a(n) = Stirling2(n, floor(n/2)) * floor(n/2)!.
%F A344397 a(n) = [x^(floor(n/2))] F(n, x), the middle coefficient of the Fubini polynomial.
%F A344397 a(n) = Sum_{k=0..n/2} (-1)^k*binomial((2*n - 1)/4 + (-1)^n/4, k)*((2*n - 1)/4 + (-1)^n/4 - k)^n.
%p A344397 a := n -> add((-1)^k*binomial((2*n-1)/4 + (-1)^n/4,k)*((2*n-1)/4 + (-1)^n/4 - k)^n, k = 0..n/2):
%p A344397 # Alternative, via Fubini recurrence:
%p A344397 F := proc(n) option remember; if n = 0 then return 1 fi;
%p A344397 expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
%p A344397 a := n -> coeff(F(n), x, iquo(n, 2));
%p A344397 seq(a(n), n = 0..22);
%t A344397 a[n_] := StirlingS2[n, Floor[n/2]] * Floor[n/2]!; Array[a, 23, 0] (* _Amiram Eldar_, May 22 2021 *)
%o A344397 (SageMath)
%o A344397 def a(n): return stirling_number2(n, n//2) * factorial(n//2)
%o A344397 print([a(n) for n in range(23)])
%Y A344397 Cf. A131689, A210029, A019538, A090582.
%K A344397 nonn
%O A344397 0,5
%A A344397 _Peter Luschny_, May 21 2021