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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.

%I A383772 #6 May 21 2025 16:38:59
%S A383772 0,-4,-18,-610,-15675,-772122,-47282844,-3918873376,-410168886615,
%T A383772 -53329052728000,-8417451284317614,-1586200451151892608,
%U A383772 -351735180091505203539,-90667510133054591492224,-26884188746929397888775000,-9086147134545912835276742656
%N 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.
%e A383772 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.
%t A383772 z = 18;
%t A383772 v[n_] := Table[k + 1, {k, 0, n - 1}];
%t A383772 u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
%t A383772 p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A085719 *)
%t A383772 d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182 *)
%t A383772 neg = (d - p)/2  (* A383772 *)
%t A383772 pos = (d + p)/2  (* A383773 *)
%Y A383772 Cf. A052182 (determinant), A085719 (permanent), A380661, A383773, A383774, A383775.
%K A383772 sign
%O A383772 1,2
%A A383772 _Clark Kimberling_, May 15 2025