A203475 a(n) = Product_{1 <= i < j <= n} (i^2 + j^2).

1, 5, 650, 5525000, 5807194900000, 1226800120038480000000, 77092420109247492627600000000000, 2001314057760220784660590245696000000000000000, 28468550112906756205383102673584071297339520000000000000000000

Offset: 1

Author

Clark Kimberling, Jan 02 2012

Keywords

• nonn

Comments

Each term divides its successor, as in A203476.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..28

Crossrefs

Cf. A000290, A093883, A202768, A203476, A272244, A293290, A323755, A324403, A367517.

Programs

Magma

[(&*[(&*[j^2 + k^2: k in [1..j]])/(2*j^2): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023

Maple

a:= n-> mul(mul(i^2+j^2, i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017

Mathematica

f[j_]:= j^2; z = 15;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]           (* A203475 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A203476 *)

SageMath

[product(product(j^2+k^2 for k in range(1,j)) for j in range(1,n+1)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023

Formula

a(n) ~ c * 2^(n^2/2) * exp(Pi*n*(n+1)/4 - 3*n^2/2 + n) * n^(n*(n-1) - 3/4), where c = A323755 = sqrt(Gamma(1/4)) * exp(Pi/24) / (2*Pi)^(9/8) = 0.274528350333552903800408993482507428142383783773190451181... - Vaclav Kotesovec, Jan 26 2019

Extensions

Name edited by Alois P. Heinz, Jul 23 2017