A306594 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i + j + k).
1, 3, 144000, 455282248974336000000, 9608917807566747651759509633033255126040576000000000000
Offset: 0
Keywords
Programs
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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
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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 *)
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.
Extensions
a(0)=1 prepended by Alois P. Heinz, Jun 24 2023
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