A324442 a(n) = Product_{i=1..n, j=1..n} (i^2 + j).

1, 2, 180, 6652800, 402265543680000, 109211487076824381849600000, 295382703175843424854047228769075200000000, 15385012566245626089929288743828190926813939944652800000000000

Offset: 0

Author

Vaclav Kotesovec, Feb 28 2019

Keywords

• nonn

Crossrefs

Cf. A079478, A101686, A255322, A324403.

Programs

Maple

a:= n-> mul(mul(i^2+j, i=1..n), j=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Jun 24 2023

Mathematica

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

Formula

From Vaclav Kotesovec, Dec 27 2023: (Start)

a(n) ~ c * n^(2*n^2 + n/2 - 1/4) / exp(2*n^2 - 2*Pi*n^(3/2)/3 - Pi*sqrt(n)/2), where c = 0.31906...

For n>1, a(n) = a(n-1) * Gamma(n - i*sqrt(n)) * Gamma(n + i*sqrt(n)) * Gamma(n^2 + n + 1) * sinh(Pi*sqrt(n)) / (Pi * n^(5/2) * Gamma(n^2)), where i is the imaginary unit. (End)

Extensions

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