cp's OEIS Frontend

In general, for d >= -1, Product_{i=1..n, j=1..n} (i^2 + d*i*j + j^2) ~ c(d) * (d+2)^((d+2)*n*(n+1)/2) * n^(2*n^2 - 1/2 - d/6) / ((d/2 + sqrt(d^2/4 - 1))^(sqrt(d^2 - 4)*n*(n+1)/2) * exp(3*n^2)), where c(d) is a constant (dependent only on d).

c(-1) = 3^(1/6) * exp(Pi/(6*sqrt(3))) * Gamma(1/3)^2 / (2*Pi)^(5/3).

c(0) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4).

c(1) = 3^(5/12) * exp(Pi/(12*sqrt(3))) * Gamma(1/3) / (2*Pi)^(4/3).

c(2) = A^2 / (2^(1/6) * exp(1/6) * Pi), where A = A074962.

c(3) = 2^((sqrt(5) - 9)/6) * sqrt(5) * (1 + sqrt(5))^(1/2 - sqrt(5)/6) / Pi.

c(4) = 2^((sqrt(3) - 1)/6) * 3^(13/24) * (1 + sqrt(3))^(1/2 - 1/sqrt(3)) / (Pi^(7/12) * Gamma(1/4)^(1/3) * Gamma(1/3)^(1/2)).

c(5) = A368069.

c(6) = 2^(25/8) * (1 + sqrt(2))^(3/4 - 2*sqrt(2)/3) / (Pi^(1/4) * Gamma(1/8) * Gamma(1/4)^(1/2)).

Special (non-integer) case: Product_{i=1..n, j=1..n} (i^2 + (d + 1/d)*i*j + j^2) ~ A^(2/d) * (Product_{j=1..d} Gamma(j/d)^(2*j/d)) * (d+1)^((d/2 + 1 + 1/(2*d))*2*n*(n+1) + (d+1)^2/(6*d) + 1/6) * n^(2*n^2 - d/6 - 1/2 - 1/(6*d)) / ((2*Pi)^((d+1)/2) * exp(3*n^2 + 1/(6*d)) * d^((d+1)*n*(n+1) - 1/(6*d))), where A = A074962 is the Glaisher-Kinkelin constant.