A384077 a(n) = neg(M(n)), where M(n) is the n X n left 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, -153, -2668, -200200, -20916552, -2515242520, -404114856640, -84096850828032, -21708790967664000, -6808856052755927808, -2552126898198385479168, -1126589571631974396251136, -578462264691449080954733568, -341831898528862885226121600000

Offset: 1

Author

Clark Kimberling, May 29 2025

Keywords

• sign

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, A384078.

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).