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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.

%I A362280 #27 Apr 25 2023 09:22:39
%S A362280 1,8,32640,606108303360,288646869784585568256000,
%T A362280 3978466023641262138239999300075520000000,
%U A362280 4808293482959682489757553576215163849442438886195200000000000,669887741948823664389458168162886859168459418141304785844082510440658108416000000000000
%N 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.
%F A362280 a(n) = A362291(n)*(m!)^2*(n^2 - 2*m)!, where m = 2*floor(n/2).
%e A362280 a(1) = A362209(1,1) = 1 since we have:
%e A362280      [1].
%e A362280 a(2) = A362209(5,2) = 8 since we have:
%e A362280      [1, 2]  [1, 3]  [4, 2]  [4, 3]
%e A362280      [3, 4], [2, 4], [3, 1], [2, 1],
%e A362280 .
%e A362280      [2, 1]  [2, 4]  [3, 1]  [3, 4]
%e A362280      [4, 3], [1, 3], [4, 2], [1, 2].
%o A362280 (Python)
%o A362280 from math import factorial
%o A362280 from itertools import combinations as C
%o A362280 def a(n):
%o A362280     E = [i for i in range(1, n**2+1)]
%o A362280     m = n if n%2 == 0 else n-1
%o A362280     r = n**2 - 2*m
%o A362280     fm, fr = factorial(m), factorial(r)
%o A362280     p = fm**2 * fr
%o A362280     return p*sum(1 for u in C(E, 2*m) for t in C(u, m) if 2*sum(t)==sum(u))
%o A362280 print([a(n) for n in range(1, 5)])
%Y A362280 Cf. A000142, A001044, A052928, A362209, A362291.
%K A362280 nonn,hard,more
%O A362280 1,2
%A A362280 _Stefano Spezia_ and _Michael S. Branicky_, Apr 14 2023
%E A362280 a(6)-a(8) calculated from A362291 by _Martin Ehrenstein_, Apr 25 2023