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

1, 1, 4, 252, 576576, 87178291200, 1386980110791475200, 3394352757964564324299571200, 1760578659300452732262852600316664217600, 255323290537547288382098619855584488593426606981120000

Offset: 0

Author

Clark Kimberling, Jan 03 2012

Keywords

• nonn

Comments

Each term divides its successor, as in A203512.

See A093883 for a guide to related sequences.

Crossrefs

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

Programs

Maple

t:= n-> n*(n+1)/2:
a:= n-> mul(mul(t(i)+t(j), i=1..j-1), j=2..n):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 23 2017

Mathematica

f[j_] := j (j + 1)/2; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]               (* A203511 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A203512 *)
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 *)

Formula

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

Extensions

Name edited by Alois P. Heinz, Jul 23 2017

a(0)=1 prepended by Alois P. Heinz, Jul 29 2017