A384075 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

0, -9, -45, -4716, -200200, -20916552, -2462535768, -406262340288, -84096850828032, -21708790967664000, -6808563893605222144, -2552145158372103507456, -1126589571631974396251136, -578462264691449080954733568, -341831891354409385226121600000

Offset: 1

Author

Clark Kimberling, May 22 2025

Keywords

• sign

Examples

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

Crossrefs

Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384077, A384078.

Programs

Mathematica

z = 19;
v[n_] := Table[2 k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, 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, with alternating signs *)
neg = (d - p)/2   (* A384075 *)
pos = (d + p)/2   (* A384076 *)

Formula

a(n) = (1/2)*((-1)^n*A193678(n) - A384074(n)).