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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.

%I A383774 #8 May 27 2025 22:11:50
%S A383774 0,-4,-36,-450,-15675,-772122,-47753440,-3909436192,-410168886615,
%T A383774 -53329052728000,-8417606908865220,-1586195621597483136,
%U A383774 -351735180091505203539,-90667510133054591492224,-26884188980472806091900000,-9086147124746080046118543360
%N 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.
%e A383774 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.
%t A383774 z = 18;
%t A383774 v[n_] := Table[k + 1, {k, 0, n - 1}];
%t A383774 u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
%t A383774 p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A085719 *)
%t A383774 d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182, with altered signs *)
%t A383774 neg = (d - p)/2   (* A383774 *)
%t A383774 pos = (d + p)/2   (* A383775 *)
%Y A383774 Cf. A052182 (determinant), A085719 (permanent), A380661, A383772, A383773, A383775.
%K A383774 sign
%O A383774 1,2
%A A383774 _Clark Kimberling_, May 17 2025