A383774 a(n) = neg(M(n)), where M(n) is the n X n left 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, -36, -450, -15675, -772122, -47753440, -3909436192, -410168886615, -53329052728000, -8417606908865220, -1586195621597483136, -351735180091505203539, -90667510133054591492224, -26884188980472806091900000, -9086147124746080046118543360

Offset: 1

Author

Clark Kimberling, May 17 2025

Keywords

• sign

Examples

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

Crossrefs

Cf. A052182 (determinant), A085719 (permanent), A380661, A383772, A383773, A383775.

Programs

Mathematica

z = 18;
v[n_] := Table[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}]   (* A085719 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182, with altered signs *)
neg = (d - p)/2   (* A383774 *)
pos = (d + p)/2   (* A383775 *)