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A203688 v(n+1)/v(n), where v=A203687.

%I A203688 #10 Nov 21 2023 08:41:00
%S A203688 3,868,137683728,33559369419469824,30814027871378630714538393600,
%T A203688 228763757339598304909903916639768291573760000,
%U A203688 26160851477909352213716196682766580796112417641624416288768000000
%N A203688 v(n+1)/v(n), where v=A203687.
%C A203688 See A093883 for a discussion and guide to related sequences.
%F A203688 From _Vaclav Kotesovec_, Nov 21 2023: (Start)
%F A203688 a(n) ~ (n+1)!^(2*n).
%F A203688 a(n) ~ (2*Pi)^n * n^(2*n^2 + 3*n) / exp(2*n^2 - 13/6). (End)
%t A203688 f[j_] := j!; z = 8;
%t A203688 u[n_] := Product[f[j]^2 - f[j] f[k] + f[k]^2,
%t A203688   {j, 1, k - 1}]
%t A203688 v[n_] := Product[u[n], {k, 2, n}]
%t A203688 Table[v[n], {n, 1, z}]          (* A203687 *)
%t A203688 Table[v[n + 1]/v[n], {n, 1, z}] (* A203688 *)
%t A203688 Table[Product[k!^2 - k!*(n+1)! + (n+1)!^2, {k, 1, n}], {n, 1, 10}] (* _Vaclav Kotesovec_, Nov 21 2023 *)
%K A203688 nonn
%O A203688 1,1
%A A203688 _Clark Kimberling_, Jan 04 2012