A203511 - OEIS

%I #23 Sep 07 2023 05:15:58

%S 1,1,4,252,576576,87178291200,1386980110791475200,

%T 3394352757964564324299571200,

%U 1760578659300452732262852600316664217600,255323290537547288382098619855584488593426606981120000

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

%C Each term divides its successor, as in A203512.

%C See A093883 for a guide to related sequences.

%F 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 t:= n-> n*(n+1)/2:

%p a:= n-> mul(mul(t(i)+t(j), i=1..j-1), j=2..n):

%p seq(a(n), n=0..12);  # _Alois P. Heinz_, Jul 23 2017

%t f[j_] := j (j + 1)/2; z = 15;

%t v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]

%t Table[v[n], {n, 1, z}]               (* A203511 *)

%t Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A203512 *)

%t 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 Cf. A000217, A203512, A293290, A324403, A324443.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jan 03 2012

%E Name edited by _Alois P. Heinz_, Jul 23 2017

%E a(0)=1 prepended by _Alois P. Heinz_, Jul 29 2017