A362280 a(n) is the number of n X n matrices using all the integers from 1 to n^2 with trace equal to the antitrace.

1, 8, 32640, 606108303360, 288646869784585568256000, 3978466023641262138239999300075520000000, 4808293482959682489757553576215163849442438886195200000000000, 669887741948823664389458168162886859168459418141304785844082510440658108416000000000000

Offset: 1

Author

Stefano Spezia and Michael S. Branicky, Apr 14 2023

Keywords

• nonn
• hard
• more

Examples

			a(1) = A362209(1,1) = 1 since we have:
     [1].
a(2) = A362209(5,2) = 8 since we have:
     [1, 2]  [1, 3]  [4, 2]  [4, 3]
     [3, 4], [2, 4], [3, 1], [2, 1],
.
     [2, 1]  [2, 4]  [3, 1]  [3, 4]
     [4, 3], [1, 3], [4, 2], [1, 2].
		

Crossrefs

Cf. A000142, A001044, A052928, A362209, A362291.

Programs

Python

from math import factorial
from itertools import combinations as C
def a(n):
    E = [i for i in range(1, n**2+1)]
    m = n if n%2 == 0 else n-1
    r = n**2 - 2*m
    fm, fr = factorial(m), factorial(r)
    p = fm**2 * fr
    return p*sum(1 for u in C(E, 2*m) for t in C(u, m) if 2*sum(t)==sum(u))
print([a(n) for n in range(1, 5)])

Formula

a(n) = A362291(n)*(m!)^2*(n^2 - 2*m)!, where m = 2*floor(n/2).

Extensions

a(6)-a(8) calculated from A362291 by Martin Ehrenstein, Apr 25 2023