A367958 - OEIS

%I #14 Dec 20 2023 08:27:27

%S 1,6,5544,2822916096,1723467782592331776,

%T 2210440498434925488635904000000,

%U 9234659938893939743399592700454853672960000000,180150216814109052335771891722360520401032374209013927116800000000

%N a(n) = Product_{i=1..n, j=1..n} (i + 5*j).

%C In general, for d>0, Product_{i=1..n, j=1..n} (i + d*j) ~ A^(1/d) * (Product_{j=1..d} Gamma(j/d)^(j/d)) * (d+1)^((d/2 + 1 + 1/(2*d))*n*(n+1) + (d+1)^2/(12*d) + 1/12) * n^(n^2 - d/12 - 1/4 - 1/(12*d)) / ((2*Pi)^((d+1)/4) * exp(3*n^2/2 + 1/(12*d)) * d^((n*(d*n + (d+1)))/2 - 1/(12*d))), where A = A074962 is the Glaisher-Kinkelin constant.

%C Equivalently, for d>0, Product_{i=1..n, j=1..n} (i + d*j) ~ A^d * (Product_{j=1..d} BarnesG(j/d)) * (2*Pi)^((d-3)/4) * (d+1)^((d + (d+1)^2*(6*n*(n+1) + 1)) / (12*d)) * n^(n^2 - 1/4 - 1/(12*d) - d/12) / (d^((n+1)*(d*n + 1)/2) * exp(3*n^2/2 + d/12)).

%F a(n) ~ A^(1/5) * (1 + sqrt(5))^(1/10) * 2^(18*n*(n+1)/5 + 29/60) * 3^(18*n*(n+1)/5 + 41/60) * n^(n^2 - 41/60) / (Pi^(1/10) * Gamma(1/5)^(3/5) * Gamma(2/5)^(1/5) * 5^(n*(5*n+6)/2 + 1/3) * exp(3*n^2/2 + 1/60)), where A = A074962 is the Glaisher-Kinkelin constant.

%p a:= n-> mul(mul(i+5*j, i=1..n), j=1..n):

%p seq(a(n), n=0..8);  # _Alois P. Heinz_, Dec 06 2023

%t Table[Product[i + 5*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

%Y Cf. A079478 (d=1), A324402 (d=2), A367956 (d=3), A367957 (d=4).

%Y Cf. A074962, A367842, A367944.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Dec 06 2023