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A203511 a(n) = Product_{1 <= i < j <= n} (t(i) + t(j)); t = A000217 = triangular numbers.

%I A203511 #23 Sep 07 2023 05:15:58
%S A203511 1,1,4,252,576576,87178291200,1386980110791475200,
%T A203511 3394352757964564324299571200,
%U A203511 1760578659300452732262852600316664217600,255323290537547288382098619855584488593426606981120000
%N A203511 a(n) = Product_{1 <= i < j <= n} (t(i) + t(j)); t = A000217 = triangular numbers.
%C A203511 Each term divides its successor, as in A203512.
%C A203511 See A093883 for a guide to related sequences.
%F A203511 a(n) ~ c * 2^n * exp(n^2*(Pi/4 - 3/2) + n*(Pi/2 + 1)) * n^(n^2 - n - 2 - Pi/8), where c = 0.2807609661547466473998991675307759198889389396430915721129636653... - _Vaclav Kotesovec_, Sep 07 2023
%p A203511 t:= n-> n*(n+1)/2:
%p A203511 a:= n-> mul(mul(t(i)+t(j), i=1..j-1), j=2..n):
%p A203511 seq(a(n), n=0..12);  # _Alois P. Heinz_, Jul 23 2017
%t A203511 f[j_] := j (j + 1)/2; z = 15;
%t A203511 v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
%t A203511 Table[v[n], {n, 1, z}]               (* A203511 *)
%t A203511 Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A203512 *)
%t A203511 Table[Product[k*(k+1)/2 + j*(j+1)/2, {k, 1, n}, {j, 1, k-1}], {n, 0, 10}] (* _Vaclav Kotesovec_, Sep 07 2023 *)
%Y A203511 Cf. A000217, A203512, A293290, A324403, A324443.
%K A203511 nonn
%O A203511 0,3
%A A203511 _Clark Kimberling_, Jan 03 2012
%E A203511 Name edited by _Alois P. Heinz_, Jul 23 2017
%E A203511 a(0)=1 prepended by _Alois P. Heinz_, Jul 29 2017