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A203512 a(n) = A203511(n+1)/A203511(n).

%I A203512 #17 Feb 23 2024 02:31:05
%S A203512 1,4,63,2288,151200,15909696,2447297356,518678754048,145022370451200,
%T A203512 51747613910720000,22956761806169786496,12397159038346976323584,
%U A203512 8008689946841913447559168,6099405371286264105062400000,5408896545253926024119820000000
%N A203512 a(n) = A203511(n+1)/A203511(n).
%H A203512 G. C. Greubel, <a href="/A203512/b203512.txt">Table of n, a(n) for n = 0..220</a>
%F A203512 a(n) ~ 2 * n^(2*n) / exp((2 - Pi/2)*n - 3*Pi/4). - _Vaclav Kotesovec_, Sep 07 2023
%t A203512 f[j_] := j (j + 1)/2; z = 15;
%t A203512 v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
%t A203512 Table[v[n], {n, 1, z}]               (* A203511 *)
%t A203512 Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A203512 *)
%t A203512 Table[Product[(n+2)*(n+1)/2 + j*(j+1)/2, {j, 1, n}], {n, 0, 10}] (* _Vaclav Kotesovec_, Sep 07 2023 *)
%o A203512 (Magma) [1] cat [(&*[(n+1)*(n+2) +j*(j+1): j in [1..n]])/2^n: n in [1..30]]; // _G. C. Greubel_, Feb 23 2024
%o A203512 (SageMath)
%o A203512 def A203512(n): return product((n+1)*(n+2)+j*(j+1) for j in range(1, n+1))//2^n
%o A203512 [A203512(n) for n in range(31)] # _G. C. Greubel_, Feb 23 2024
%Y A203512 Cf. A000217, A093883, A203511.
%K A203512 nonn
%O A203512 0,2
%A A203512 _Clark Kimberling_, Jan 03 2012
%E A203512 More terms from _Alois P. Heinz_, Jul 29 2017