A367517 -id:A367517 - cp's OEIS frontend

A324426 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3).

1, 2, 2592, 134425267200, 3120795915109442519040000, 180825857777547616919759624941086965760000000, 99356698720512072045648926659510730227553351200000695922065408000000000

Offset: 0

Vaclav Kotesovec, Feb 27 2019

nonn

Cf. A079478, A324403, A324437, A324438, A324439, A324440, A367834.

Cf. A272246, A307210, A367517, A367543.

a:= n-> mul(mul(i^3+j^3, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^3+j^3, {i, 1, n}, {j, 1, n}], {n, 1, 10}]

a(n) = prod(i=1, n, prod(j=1, n, i^3+j^3)); \\ Michel Marcus, Feb 27 2019

from math import prod, factorial
def A324426(n): return prod(i**3+j**3 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**3<Chai Wah Wu, Nov 26 2023

a(n) ~ A * 2^(2*n*(n+1) + 1/4) * exp(Pi*(n*(n+1) + 1/6)/sqrt(3) - 9*n^2/2 - 1/12) * n^(3*n^2 - 3/4) / (3^(5/6) * Pi^(1/6) * Gamma(2/3)^2), where A is the Glaisher-Kinkelin constant A074962.
a(n) = A079478(n) * A367543(n). - Vaclav Kotesovec, Nov 22 2023
For n>0, a(n)/a(n-1) = A272246(n)^2 / (2*n^9). - Vaclav Kotesovec, Dec 02 2023

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

A203475 a(n) = Product_{1 <= i < j <= n} (i^2 + j^2).

Original entry on oeis.org

1, 5, 650, 5525000, 5807194900000, 1226800120038480000000, 77092420109247492627600000000000, 2001314057760220784660590245696000000000000000, 28468550112906756205383102673584071297339520000000000000000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A203476.

G. C. Greubel, Table of n, a(n) for n = 1..28

Cf. A000290, A093883, A202768, A203476, A272244, A293290, A323755, A324403, A367517.

[(&*[(&*[j^2 + k^2: k in [1..j]])/(2*j^2): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023

a:= n-> mul(mul(i^2+j^2, i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017

f[j_]:= j^2; z = 15;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]           (* A203475 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A203476 *)

[product(product(j^2+k^2 for k in range(1,j)) for j in range(1,n+1)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023

a(n) ~ c * 2^(n^2/2) * exp(Pi*n*(n+1)/4 - 3*n^2/2 + n) * n^(n*(n-1) - 3/4), where c = A323755 = sqrt(Gamma(1/4)) * exp(Pi/24) / (2*Pi)^(9/8) = 0.274528350333552903800408993482507428142383783773190451181... - Vaclav Kotesovec, Jan 26 2019

Name edited by Alois P. Heinz, Jul 23 2017

cp's OEIS Frontend

A324426 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3).

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