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A384313 a(n) = pos(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 pos(M(n)) is the positive part of the determinant of M(n); see A380661.

%I A384313 #7 Jul 02 2025 00:52:42
%S A384313 0,0,2,9,582,27136,7661772,2797055478,4374706319136,11681281664592429,
%T A384313 112352959301265272414,2147474541377915674682880,
%U A384313 133430162305143400794479937840,18069411470335957872130103264497774,7436752857750595469877425837627133763584
%N A384313 a(n) = pos(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 pos(M(n)) is the positive part of the determinant of M(n); see A380661.
%F A384313 a(n) = (-(-1)^n * A123744(n) + A384079(n)) / 2.
%e A384313 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 A384313 z = 14;
%t A384313 v[n_] := Table[Fibonacci[k], {k, 0, n - 1}];
%t A384313 u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
%t A384313 p = Table[Permanent[u[n]], {n, 1, z}]   (* A384079 *)
%t A384313 d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123744 *)
%t A384313 neg = (d - p)/2    (* A384080 *)
%t A384313 pos = (d + p)/2    (* A384313 *)
%Y A384313 Cf. A123744 (determinant), A384079 (permanent), A380661, A384080.
%K A384313 nonn
%O A384313 1,3
%A A384313 _Clark Kimberling_, Jun 27 2025