A384078 a(n) = pos(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

1, 1, 45, 4716, 250200, 19423560, 2462535768, 406262340288, 84196030473216, 21703670967664000, 6808563893605222144, 2552145158372103507456, 1126590812208410998119424, 578462173661889165983466496, 341831891354409385226121600000

Offset: 1

Author

Clark Kimberling, Jun 01 2025

Keywords

• nonn

Examples

			The rows of M(4) are (1,3,5,7), (3,5,7,1), (5,7,1,3), (7,1,3,5); determinant(M(4)) = 2048; permanent(M(4)) = 7384, so neg(M(4)) = (7384 - 2048)/2 = -2668 and pos(M(4)) = (7384+2048)/2 = 4716.
		

Crossrefs

Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384077.

Programs

Mathematica

z = 15;
v[n_] := Table[2 k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]     (* A384074 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A193678 up to signs *)
neg = (d - p)/2    (* A384077 *)
pos = (d + p)/2    (* A384078 *)

Formula

a(n) = (1/2)*(s(n)*A193678(n) - A384074(n)), where s(n) = (-1)^((2*n+(-1)^n-1)/4).