A306620 -id:A306620 - cp's OEIS frontend

A324437 a(n) = Product_{i=1..n, j=1..n} (i^4 + j^4).

1, 2, 18496, 189567553208832, 53863903330477722171391434817536, 4194051697335929481600368256016484482740174637152337920000, 530545265060440849231458462212366841894726534233233018777709463062563850450708386692464640000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Cf. A079478, A324403, A324426, A324438, A324439, A324440, A367834.

Cf. A203677, A272247, A307215.

a:= n-> mul(mul(i^4+j^4, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^4 + j^4, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

from math import prod, factorial
def A324437(n): return (prod(i**4+j**4 for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**2)**2<Chai Wah Wu, Nov 26 2023

a(n) ~ c * 2^(n*(n+1)) * exp(Pi*n*(n+1)/sqrt(2) - 6*n^2) * (1 + sqrt(2))^(sqrt(2)*n*(n+1)) * n^(4*n^2 - 1), where c = A306620 = 0.23451584451404279281807143317500518660696293944961...

For n>0, a(n)/a(n-1) = A272247(n)^2 / (2*n^12). - Vaclav Kotesovec, Dec 01 2023

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

A367834 a(n) = Product_{i=1..n, j=1..n} (i^8 + j^8).

Original entry on oeis.org

1, 2, 67634176, 1775927682136440882473213952, 22495149450984565292579847926810488282934424886723006835982336

Offset: 0

Vaclav Kotesovec, Dec 02 2023

nonn

Next term is too long to be included.
For m > 0, Product_{j=1..n, k=1..n} (j^m + k^m) ~ c(m) * exp(n*(n+1)*s(m) - m*n*(n-2)/2) * n^(m*(n^2 - 1/4 - v)), where v = 0 if m > 1 and v = 1/6 if m = 1, s(m) = Sum_{j>=1} (-1)^(j+1)/(j*(1 + m*j)) and c(m) is a constant (dependent only on m). Equivalently, s(m) = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/m).
c(1) = A / (2^(1/12) * exp(1/12) * sqrt(Pi)).
c(2) = exp(Pi/12) * Gamma(1/4) / (2^(5/4) * Pi^(5/4)).
c(3) = A * 3^(1/6) * exp(Pi/(6*sqrt(3)) - 1/12) * Gamma(1/3)^2 / (2^(7/4) * Pi^(13/6)), where A = A074962 is the Glaisher-Kinkelin constant.
c(4) = A306620.

Cf. A079478 (m=1), A324403 (m=2), A324426 (m=3), A324437 (m=4), A324438 (m=5), A324439 (m=6), A324440 (m=7).

Cf. A306620, A367670, A367833.

Table[Product[i^8 + j^8, {i, 1, n}, {j, 1, n}], {n, 0, 6}]

from math import prod, factorial
def A367834(n): return (prod(i**8+j**8 for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**4)**2<Chai Wah Wu, Dec 02 2023

For n>0, a(n)/a(n-1) = A367833(n)^2 / (2*n^24).

a(n) ~ c * 2^(n*(n+1)) * (1 + 1/(sqrt(1 - 1/sqrt(2)) - 1/2))^(sqrt(2 + sqrt(2))*n*((n+1)/2)) * (1 + 1/(sqrt(1 + 1/sqrt(2)) - 1/2))^(sqrt(2 - sqrt(2))*n*((n+1)/2)) * (n^(8*n^2 - 2) / exp(12*n^2 - Pi*sqrt(1 + 1/sqrt(2))*n*(n+1))), where c = 0.043985703178712025347328240881106818917398444790454628282522057393529338998...

cp's OEIS Frontend

A324437 a(n) = Product_{i=1..n, j=1..n} (i^4 + j^4).

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