%I #23 Jan 06 2024 14:29:45

%S 1,3,144000,455282248974336000000,

%T 9608917807566747651759509633033255126040576000000000000

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

%C Next term is too long to be included.

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

%F 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)).

%F 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.

%p a:= n-> mul(mul(mul(i+j+k, i=1..n), j=1..n), k=1..n):

%p seq(a(n), n=0..5); # _Alois P. Heinz_, Jun 24 2023

%t Table[Product[i+j+k, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 6}]

%t 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}]

%t 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 *)

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

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Feb 27 2019

%E a(0)=1 prepended by _Alois P. Heinz_, Jun 24 2023