A368066 a(n) = Product_{i=1..n, j=1..n} (i^2 + 6*i*j + j^2).

1, 8, 73984, 10027173445632, 93867986947606492024406016, 185865459466664040069739311383413462872883200, 186896871826703385639703785281909582209471190408233074664996759142400

Offset: 0

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

• nonn

Comments

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.

Crossrefs

Cf. A367543 (d=-1), A324403 (d=0), A367542 (d=1), A079478^2 (d=2), A368067 (d=3), A368064 (d=4), A368065 (d=5).

Cf. A368069, A368068.

Programs

Mathematica

Table[Product[i^2 + 6*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) ~ 2^(12*n*(n+1) + 25/8) * n^(2*n^2 - 3/2) / (Pi^(1/4) * Gamma(1/4)^(1/2) * Gamma(1/8) * (1 + sqrt(2))^(2*sqrt(2)*(6*n*(n+1) + 1)/3 - 3/4) * exp(3*n^2)).