A383772 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1, 2, ... , n), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

0, -4, -18, -610, -15675, -772122, -47282844, -3918873376, -410168886615, -53329052728000, -8417451284317614, -1586200451151892608, -351735180091505203539, -90667510133054591492224, -26884188746929397888775000, -9086147134545912835276742656

Offset: 1

Author

Clark Kimberling, May 15 2025

Keywords

• sign

Examples

			The rows of M(4) are (1, 2, 3, 4), (4, 1, 2, 3), (3, 4, 1, 2), (2, 3, 4, 1); determinant(M(4)) = -160; permanent(M(4)) = 1060, so neg(M(4)) = (-160 - 1060)/2 = -610 and pos(M(4)) = (-160 + 1060)/2 = 450.
		

Crossrefs

Cf. A052182 (determinant), A085719 (permanent), A380661, A383773, A383774, A383775.

Programs

Mathematica

z = 18;
v[n_] := Table[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}]   (* A085719 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182 *)
neg = (d - p)/2  (* A383772 *)
pos = (d + p)/2  (* A383773 *)