A306594 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i + j + k).

1, 3, 144000, 455282248974336000000, 9608917807566747651759509633033255126040576000000000000

Offset

0,2

Comments

Next term is too long to be included.

Links

Table of n, a(n) for n=0..4.

Formula

a(n) = Product_{k=1..n} (BarnesG(k+2) * BarnesG(2*n+k+2) / BarnesG(n+k+2)^2).

a(n) = Product_{k=1..n} (k^(3*(n - k + 1)*(n - k + 2)/2)) * Product_{k=1..3*n} (k^((3*n - k + 1)*(3*n - k + 2)/2)) / Product_{k=1..2*n} (k^(3*(2*n - k + 1)*(2*n - k + 2)/2)).

a(n) ~ sqrt(Pi) * 3^(9*n^3/2 + 27*n^2/4 + 3*n + 3/8) * n^(n^3 + 3/8) / (A^(3/2) * 2^(4*n^3 + 9*n^2 + 6*n + 5/8) * exp(11*n^3/6 - Zeta(3)/(8*Pi^2) - 1/8)), where A is the Glaisher-Kinkelin constant A074962.

Maple

a:= n-> mul(mul(mul(i+j+k, i=1..n), j=1..n), k=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Jun 24 2023

Mathematica

Table[Product[i+j+k, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 6}]
Table[Product[k^(3*(n - k + 1) (n - k + 2)/2), {k, 1, n}] * Product[k^((3*n - k + 1) (3*n - k + 2)/2), {k, 1, 3*n}] / Product[k^(3*(2*n - k + 1) (2*n - k + 2)/2), {k, 1, 2*n}], {n, 1, 6}]
Clear[a]; a[n_] := a[n] = If[n == 1, 3, 3*n*a[n-1] * BarnesG[2+n]^3 * BarnesG[2+3*n]^3 * Gamma[1+2*n]^3 / (BarnesG[2+2*n]^6 * Gamma[1+3*n]^3)]; Table[a[n], {n, 1, 6}] (* Vaclav Kotesovec, Mar 28 2019 *)

Crossrefs

Cf. A079478, A093884, A112332, A324425, A368722, A368723, A324441.

Sequence in context: A116536 A224241 A178505 * A003544 A250495 A317168

Adjacent sequences: A306591 A306592 A306593 * A306595 A306596 A306597

Keyword

nonn

Author

Vaclav Kotesovec, Feb 27 2019

Extensions

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

Status

approved