A307209 -id:A307209 - cp's OEIS frontend

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

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

A307215 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^4 + j^4)).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 29 2019

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...
Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)) = A307209 = 3.50478299933972837589112...

			1.94073028537236152995386077599647772038707968293217092813061397472522642172...

Cf. A307209, A324437.

(* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^4 + j^4), {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}]

Equals limit_{n->infinity} (Product_{i=1..n, j=1..n} (1 + i^4 + j^4)) / A324437(n).

cp's OEIS Frontend

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

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