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A384593 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

%I A384593 #4 Jul 03 2025 16:22:01
%S A384593 1,1,10,115,7182,630400,236525784,146492200737,343945864714624,
%T A384593 1504008427530969535,22752527891269251753118,
%U A384593 704885020930148344661403136,70113811460119152088048405685808,15355281759224121782879344275585373609
%N A384593 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.
%F A384593 a(n) = (1/2)*((-1)^n A123745(n) + A384591(n)).
%e A384593 The rows of M(4) are (1,1,2,3), (3,1,1,2), (2,3,1,1), (1,2,3,1); determinant(M(4)) = -35; permanent(M(4)) = 265, so neg(M(4)) = (-35 - 265)/2 = -150 and pos(M(4)) = (-35 + 265)/2 = 115.
%t A384593 z = 14;
%t A384593 v[n_] := Table[Fibonacci[k], {k, 1, n}];
%t A384593 u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
%t A384593 Table[Permanent[u[n]], {n, 1, z}]   (* A384591  *)
%t A384593 d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123745, with alternating signs *)
%t A384593 neg = (d - p)/2     (* A384592 *)
%t A384593 pos = (d + p)/2     (* A384593 *)
%Y A384593 Cf. A000045, A380661, A123745 (determinant), A384591 (permanent), A384592.
%K A384593 nonn
%O A384593 1,3
%A A384593 _Clark Kimberling_, Jul 02 2025