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.

1, 4, 225, 27728, 7240350, 3439197360, 2686774125000, 3213645578293248, 5578750547986764960, 13484491722080225280000, 43904082301794970311672000, 187409206411313292409598115840, 1025421491750171253824589270768000, 7056011383804251291488039375527526400

Offset: 1

Author

Clark Kimberling, Mar 09 2025

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A232773, A380661, A380662.

Programs

Mathematica

r[m_, n_] := Range[(m - 1) n + 1, m  n];
d = Table[Det[Table[r[m, n], {m, 1, n}]], {n, 1, 15}]
p = Table[Permanent[Table[r[m, n], {m, 1, n}]], {n, 1, 15}]
neg = (d - p)/2
pos = (d + p)/2

Python

from sympy.functions.combinatorial.numbers import stirling, factorial
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

Formula

a(n) = A232773(n)/2 for n >= 3.