A307215 -id:A307215 - 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

A307209 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)).

Original entry on oeis.org

3, 5, 0, 4, 7, 8, 2, 9, 9, 9, 3, 3, 9, 7, 2, 8, 3, 7, 5, 8, 9, 1, 1, 2, 0, 5, 7, 0, 4, 3, 8, 0, 6, 1, 2, 5, 5, 8, 3, 8, 9, 3, 2, 4, 7, 8, 6, 2, 7, 1, 2, 7, 5, 3, 5, 4, 1, 9, 9, 4, 6, 2, 6, 6, 1, 4, 0, 5, 8, 3, 8, 5, 0, 3, 5, 0, 3, 4, 7, 5, 6, 3, 5, 2, 7, 4, 7, 5, 0, 9, 5, 0, 5, 1, 3, 7, 8, 9, 1, 7, 8, 4, 5, 9, 7

Offset: 1

Vaclav Kotesovec, Mar 28 2019

Product_{i>=1, j>=1} (1 + 1/(i^2 + j^2)) is divergent.
A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313...
Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).

			3.50478299933972837589112057043806125583893247862712753541994626614058385...

Vaclav Kotesovec, Table of n, a(n) for n = 1..224

Cf. A073017, A307210, A307215, A324403, A324426, A324443.

(* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^3 + j^3), {i, 1, n}, {j, 1, n}]; Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 100]], {m, 10, 100, 10}]

default(realprecision, 50); exp(sumalt(k=1, -(-1)^k/k*sumnum(i=1, sumnum(j=1, 1/(i^3+j^3)^k)))) \\ 15 decimals correct

Equals limit_{n->infinity} A307210(n) / A324426(n).

cp's OEIS Frontend

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

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A307209 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)).

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