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a(n) = Product_{i=1..n, j=1..n} (i^2 + j^2).
34

%I #34 Jan 01 2024 05:38:56

%S 1,2,400,121680000,281324160000000000,

%T 15539794609114833408000000000000,

%U 49933566483104048708063697937367040000000000000000,19323883089768863178599626514889213871887405416448000000000000000000000000

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

%C Next term is too long to be included.

%F 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)).

%F a(n) = 2*n^2*a(n-1)*Product_{i=1..n-1} (n^2 + i^2)^2. - _Chai Wah Wu_, Feb 26 2019

%F For n>0, a(n)/a(n-1) = A272244(n)^2 / (2*n^6). - _Vaclav Kotesovec_, Dec 02 2023

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

%p seq(a(n), n=0..7); # _Alois P. Heinz_, Jun 24 2023

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

%o (PARI) a(n) = prod(i=1, n, prod(j=1, n, i^2+j^2)); \\ _Michel Marcus_, Feb 27 2019

%o (Python)

%o from math import prod, factorial

%o 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<<n # _Chai Wah Wu_, Nov 22 2023

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

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

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Feb 26 2019

%E a(0)=1 prepended by _Alois P. Heinz_, Jun 24 2023