A324426 -id:A324426 - cp's OEIS frontend

A367517 a(n) = Product_{1 <= i < j <= n} (i^3 + j^3).

1, 9, 8820, 3756261600, 1808353823416358400, 2039434890206782239939575808000, 9701283544768145414379038964290421034844160000, 318565016660642285381616434022976430918222575100233223503872000000

Offset: 1

Vaclav Kotesovec, Nov 21 2023

nonn

Cf. A203475, A324426.

Cf. A093883, A203312.

Table[Product[Product[i^3 + j^3, {i,1,j-1}], {j,2,n}], {n,1,10}]

from math import prod
def A367517(n): return prod(i**3+j**3 for i in range(1,n) for j in range(i+1,n+1)) # Chai Wah Wu, Nov 22 2023

a(n) ~ c * 2^(n^2 + n/2) * n^(3*n*(n-1)/2 - 9/8) / exp(9*n^2/4 - Pi*n*(n+1) / (2*sqrt(3)) - 3*n/2), where c = 0.1474592465511237361695489773766942970580292170583524962797073759996256...

a(n) = A093883(n) * A203312(n).

A368720 a(n) = Product_{j=1..n, k=1..n} (j^3 + k^3 + n^3).

Original entry on oeis.org

1, 3, 69360, 1522177267723200, 391047538356893112890665992192000, 6785985430272886334677590546861463643140253138288640000000

Offset: 0

Vaclav Kotesovec, Jan 04 2024

nonn

Cf. A368685, A368622, A368721.

Cf. A324426, A368722.

Table[Product[j^3 + k^3 + n^3, {j, 1, n}, {k, 1, n}], {n, 0, 6}]

Limit_{n->oo} a(n)^(1/(n^2)) / n^3 = exp(Integral_{x=0..1, y=0..1} log(x^3 + y^3 + 1) dy dx) = 1.4501318426779115107161463152054835452511102493859148975306452...

A307210 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3 + 1).

Original entry on oeis.org

1, 3, 5100, 305727048000, 7748770873210669158912000, 476007332700693200670745550306381336371200000, 272661655519533773844144991586798737775635133552905539740860416000000000

Offset: 0

Vaclav Kotesovec, Mar 28 2019

nonn

Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).

Product_{i=1..n, j=1..n} (1 + 1/(i^2 + j^2)) = A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313315993144237973600...

Cf. A307209, A324403, A324426, A324443, A324444.

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

Table[Product[i^3 + j^3 + 1, {i, 1, n}, {j, 1, n}], {n, 1, 8}]

a(n) ~ A307209 * A324426(n).

a(n) ~ c * 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 c = A307209 = Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)) = 3.504782999339728375891120570... and A is the Glaisher-Kinkelin constant A074962.

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

cp's OEIS Frontend

A367517 a(n) = Product_{1 <= i < j <= n} (i^3 + j^3).

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A368720 a(n) = Product_{j=1..n, k=1..n} (j^3 + k^3 + n^3).

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A307210 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3 + 1).

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