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A381723 a(n) = pos(M(n)), where M(n) is the n X n matrix with numbers 1, 2, ..., n^2 in order across rows, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

%I A381723 #17 Mar 26 2025 08:28:14
%S A381723 1,4,225,27728,7240350,3439197360,2686774125000,3213645578293248,
%T A381723 5578750547986764960,13484491722080225280000,
%U A381723 43904082301794970311672000,187409206411313292409598115840,1025421491750171253824589270768000,7056011383804251291488039375527526400
%N A381723 a(n) = pos(M(n)), where M(n) is the n X n matrix with numbers 1, 2, ..., n^2 in order across rows, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.
%F A381723 a(n) = A232773(n)/2 for n >= 3.
%e A381723 M(3) is the matrix with rows (1,2,3), (4,5,6), (7,8,9), determinant 0, permanent 450, negative part -225, and positive part 225.
%t A381723 r[m_, n_] := Range[(m - 1) n + 1, m  n];
%t A381723 d = Table[Det[Table[r[m, n], {m, 1, n}]], {n, 1, 15}]
%t A381723 p = Table[Permanent[Table[r[m, n], {m, 1, n}]], {n, 1, 15}]
%t A381723 neg = (d - p)/2
%t A381723 pos = (d + p)/2
%o A381723 (Python)
%o A381723 from sympy.functions.combinatorial.numbers import stirling, factorial
%o A381723 def A381723(n): return abs(sum(n**k*stirling(n,n-k,kind=1,signed=True)*stirling(n+1,k+1,kind=1,signed=True)*factorial(n-k)*factorial(k) for k in range(n+1)))>>1 if n>2 else 3*n-2 # _Chai Wah Wu_, Mar 25 2025
%Y A381723 Cf. A232773, A380661, A380662.
%K A381723 nonn
%O A381723 1,2
%A A381723 _Clark Kimberling_, Mar 09 2025