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

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

Offset

0,2

Comments

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...

Links

Table of n, a(n) for n=0..6.

Formula

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.

Maple

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

Mathematica

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

Crossrefs

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

Sequence in context: A251603 A168556 A200950 * A171362 A249511 A362099

Adjacent sequences: A307207 A307208 A307209 * A307211 A307212 A307213

Keyword

nonn

Author

Vaclav Kotesovec, Mar 28 2019

Extensions

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

Status

approved