A324403 -id:A324403 - cp's OEIS frontend

A325048 a(n) = Product_{i=0..n, j=0..n} (i!^2 + j!^2).

2, 16, 80000, 17272267776000000, 277884245560378426290863196025651200000000, 3337940951837185557810120427617693521487357301121536848574225250643001642844160000000000

Offset: 0

Vaclav Kotesovec, Mar 26 2019

nonn

Cf. A306729, A324403, A325050.

Table[Product[i!^2 + j!^2, {i, 0, n}, {j, 0, n}], {n, 0, 7}]
Clear[a]; a[n_] := a[n] = If[n == 0, 2, a[n-1] * Product[k!^2 + n!^2, {k, 0, n}]^2 / (2*n!^2)]; Table[a[n], {n, 0, 7}]

from math import prod, factorial as f
def a(n): return prod(f(i)**2+f(j)**2 for i in range(n) for j in range(n))
print([a(n) for n in range(1, 7)]) # Michael S. Branicky, Feb 16 2021

a(n) ~ c * 2^(n*(n+3)) * Pi^(n*(n+2)) * n^((n+1)*(2*n+1)*(2*n+3)/3) / exp(2*n*(2*n+3)*(4*n+3)/9), where c = 401.488675138779168689540247334821476110398137334270208637438...

A368623 a(n) = Product_{k=1..n} (k^2 + 2*n^2).

Original entry on oeis.org

1, 3, 108, 11286, 2337984, 804305700, 414285404544, 298436020283016, 286455044544970752, 353358684943164351792, 544692796454778554880000, 1025983872949208210500475232, 2318663822077115453077590638592, 6191980828123077577798830642106944, 19289639610614384872295428226588737536

Offset: 0

Vaclav Kotesovec, Jan 01 2024

nonn

In general, for d>0, Product_{k=1..n} (k^2 + d*n^2) ~ (d+1)^(n + 1/2) * exp(n*(sqrt(d)*(Pi - 2*arctan(sqrt(d))) - 2)) * n^(2*n) / sqrt(d). - Vaclav Kotesovec, Jan 06 2024

Cf. A272244, A324403, A324425.

Table[Product[k^2 + 2*n^2, {k, 1, n}], {n, 0, 20}]

a(n) ~ 3^(n + 1/2) * exp(n*(sqrt(2)*arctan(2*sqrt(2)) - 2)) * n^(2*n) / sqrt(2).

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

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Cf. A079478, A101686, A255322, A324403.

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

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}]

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)

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

cp's OEIS Frontend

A325048 a(n) = Product_{i=0..n, j=0..n} (i!^2 + j!^2).

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A368623 a(n) = Product_{k=1..n} (k^2 + 2*n^2).

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

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