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A203690 v(n+1)/v(n), where v=A203689.

%I A203690 #10 Nov 21 2023 04:28:04
%S A203690 5,1480,204811920,44845458424326144,38524837563190678680163123200,
%T A203690 274281830629678279850073754564550983680000000,
%U A203690 30490404582529065278831553253825833486757227887272069693440000000
%N A203690 v(n+1)/v(n), where v=A203689.
%C A203690 See A093883 for a discussion and guide to related sequences.
%F A203690 From _Vaclav Kotesovec_, Nov 21 2023: (Start)
%F A203690 a(n) ~ (n+1)!^(2*n).
%F A203690 a(n) ~ (2*Pi)^n * n^(2*n^2 + 3*n) / exp(2*n^2 - 13/6). (End)
%t A203690 f[j_] := j!; z = 8;
%t A203690 u[n_] := Product[f[j]^2 + f[k]^2, {j, 1, k - 1}]
%t A203690 v[n_] := Product[u[n], {k, 2, n}]
%t A203690 Table[v[n], {n, 1, z}]          (* A203689 *)
%t A203690 Table[v[n + 1]/v[n], {n, 1, z}] (* A203690 *)
%t A203690 Table[Product[k!^2 + (n+1)!^2, {k, 1, n}], {n, 1, 10}] (* _Vaclav Kotesovec_, Nov 21 2023 *)
%K A203690 nonn
%O A203690 1,1
%A A203690 _Clark Kimberling_, Jan 04 2012