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a(n) = Product_{j=1..n, k=1..n} (j + k + n).
3

%I #4 Jan 03 2024 03:40:24

%S 1,3,600,35562240,1434015830016000,70448433354492434841600000,

%T 6610702315560389323908439364075520000000,

%U 1709479709147705756603303596364188306401499545600000000000,1660017838341811463102474357555838707949172571314554168163386261504000000000000

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

%F For n>0, a(n) = 3*BarnesG(n) * BarnesG(3*n) * Gamma(n)^2 * Gamma(3*n)^2 / (4*BarnesG(2*n)^2 * Gamma(2*n)^4).

%F a(n) ~ 3^(9*n^2/2 + 3*n + 5/12) * n^(n^2) / (2^(4*n^2 + 4*n + 5/6) * exp(3*n^2/2)).

%t Table[Product[i+j+n, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

%t Join[{1}, Table[3*BarnesG[n] * BarnesG[3*n] * Gamma[n]^2 * Gamma[3*n]^2 / (4*BarnesG[2*n]^2 * Gamma[2*n]^4), {n, 1, 8}]]

%Y Cf. A079478, A306594, A324444, A368622, A368686.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jan 03 2024