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A384080 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(0), F(1), ..., 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 A384080 #7 Jul 02 2025 00:53:55
%S A384080 0,-1,0,-25,-295,-43264,-5469632,-3628008315,-3569061677472,
%T A384080 -13761972434293885,-98350155131379362607,-2395228216526569309464064,
%U A384080 -121960521137098218596500559704,-19460957348767631231695727354978359,-6994735829985160817748505807288716492800
%N A384080 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(0), F(1), ..., 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 A384080 a(n) = (-(-1)^n * A123744(n) - A384079(n)) / 2.
%e A384080 The rows of M(4) are (0,1,1,2), (2,0,1,1), (1,2,0,1), (1,1,2,0); determinant(M(4)) = -16; permanent(M(4)) = 34, so neg(M(4)) = (-16 - 34)/2 = -25 and pos(M(4)) = (-16 + 34)/2 = 9.
%t A384080 z = 14;
%t A384080 v[n_] := Table[Fibonacci[k], {k, 0, n - 1}];
%t A384080 u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
%t A384080 p = Table[Permanent[u[n]], {n, 1, z}]   (* A384079 *)
%t A384080 d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123744, with alternating signs *)
%t A384080 neg = (d - p)/2    (* A384080 *)
%t A384080 pos = (d + p)/2    (* A384313 *)
%Y A384080 Cf. A123744 (determinant), A384079 (permanent), A380661, A384313.
%K A384080 sign
%O A384080 1,4
%A A384080 _Clark Kimberling_, Jun 19 2025