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A384595 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(2), F(3), ..., F(n+1)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

%I A384595 #4 Jul 14 2025 23:55:43
%S A384595 0,-4,-18,-1059,-51115,-14122480,-5176201331,-8184762199782,
%T A384595 -21582120875577408,-211126151053299550639,-3968236858233834575013603,
%U A384595 -250193703665647266489840668160,-33362066597786815040358189976876663,-13879811335315653909400110618024123820786
%N A384595 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(2), F(3), ..., F(n+1)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.
%F A384595 a(n) = (1/2)*(A118704(n) - A384594(n)).
%e A384595 The rows of M(4) are (1,2,3,5), (5,1,2,3), (3,5,1,2), (2,3,5,1); determinant(M(4)) = -429; permanent(M(4)) = 1689, so neg(M(4)) = (-429 - 1689)/2 = -1059 and pos(M(4)) = (-429 + 1689)/2 = 630.
%t A384595 z = 14;
%t A384595 v[n_] := Table[Fibonacci[k], {k, 2, n + 1}];
%t A384595 u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
%t A384595 p = Table[Permanent[u[n]], {n, 1, z}]   (* A384594 *)
%t A384595 d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A118704 *)
%t A384595 neg = (d - p)/2     (* A384595 *)
%t A384595 pos = (d + p)/2     (* A384596 *)
%Y A384595 Cf. A000045, A118704 (determinant), A384594 (permanent), A384596.
%K A384595 sign
%O A384595 1,2
%A A384595 _Clark Kimberling_, Jul 10 2025