A324403 a(n) = Product_{i=1..n, j=1..n} (i^2 + j^2).

1, 2, 400, 121680000, 281324160000000000, 15539794609114833408000000000000, 49933566483104048708063697937367040000000000000000, 19323883089768863178599626514889213871887405416448000000000000000000000000

Offset: 0

Author

Vaclav Kotesovec, Feb 26 2019

Keywords

• nonn

Comments

Next term is too long to be included.

Crossrefs

Cf. A079478, A324426, A324437, A324438, A324439, A324440, A367834.

Cf. A272244, A293290, A324402, A324443, A324425.

Programs

Maple

a:= n-> mul(mul(i^2+j^2, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Mathematica

Table[Product[i^2+j^2, {i, 1, n}, {j, 1, n}], {n, 1, 10}]

PARI

a(n) = prod(i=1, n, prod(j=1, n, i^2+j^2)); \\ Michel Marcus, Feb 27 2019

Python

from math import prod, factorial
def A324403(n): return (prod(i**2+j**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2<Chai Wah Wu, Nov 22 2023

Formula

a(n) ~ 2^(n*(n+1) - 3/4) * exp(Pi*n*(n+1)/2 - 3*n^2 + Pi/12) * n^(2*n^2 - 1/2) / (Pi^(1/4) * Gamma(3/4)).

a(n) = 2*n^2*a(n-1)*Product_{i=1..n-1} (n^2 + i^2)^2. - Chai Wah Wu, Feb 26 2019

For n>0, a(n)/a(n-1) = A272244(n)^2 / (2*n^6). - Vaclav Kotesovec, Dec 02 2023

a(n) = exp(2*Integral_{x=0..oo} (n^2/(x*exp(x)) - (cosh(n*x) - cos(n*x))/(x*exp((n + 1)*x)*(cosh(x) - cos(x)))) dx)/2^(n^2). - Velin Yanev, Jun 30 2025

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023