A367944 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*j^2).

1, 6, 27216, 1344924798336, 3605580335899213007486976, 1648055031941075082958467426002632704000000, 312704667066499295437237787452750428210311485710262201221120000000

Offset: 0

Author

Vaclav Kotesovec, Dec 05 2023

Keywords

• nonn

Comments

In general, for d>0, Product_{i=1..n, j=1..n} (i^2 + d*j^2) ~ c(d) * n^(2*n^2 - 1/2) * (d+1)^(n*(n+1)) * d^(-n/2) * exp(n*(n+1)*(Pi*d/2 - (d-1)*arctan(sqrt(d))) / sqrt(d) - 3*n^2), where c(d) is a constant (dependent only on d).

c(1) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4), cf. A324403.

Crossrefs

Cf. A324403 (d=1), A367941 (d=2), A367942 (d=3), A367943 (d=4).

Cf. A367542, A367543, A367958.

Programs

Mathematica

Table[Product[i^2+5*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

Formula

a(n) ~ c * n^(2*n^2 - 1/2) * 6^(n*(n+1)) * 5^(-n/2) * exp(n*(n+1)*(5*Pi/2 - 4*arctan(sqrt(5)))/sqrt(5) - 3*n^2), where c = 0.4431081869167792949266065295798218232844989957987096447783995373751372668...