A367542 a(n) = Product_{i=1..n, j=1..n} (i^2 + i*j + j^2).

3, 1764, 2905736652, 66016970246853190656, 64657853715047202043531429875379200, 6627957368676918780503749855130249245999452089509478400

Offset: 1

Author

Vaclav Kotesovec, Nov 22 2023

Keywords

• nonn

Crossrefs

Cf. A203012, A324403, A367543.

Programs

Mathematica

Table[Product[Product[(i^2 + i*j + j^2), {i, 1, n}], {j, 1, n}], {n, 1, 10}]

Python

from math import prod, factorial
def A367542(n): return (prod(i*(i+j)+j**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2*3**n # Chai Wah Wu, Nov 22 2023

Formula

a(n) ~ c * 3^(3*n*(n+1)/2) * n^(2*n^2 - 2/3) / exp(3*n^2 - Pi*n*(n+1) / (2*sqrt(3))), where c = 3^(5/12) * exp(Pi/(12*sqrt(3))) * Gamma(1/3) / (2^(4/3) * Pi^(4/3)) = 0.42478290981890921418850643030484274341562970375995548434917...