cp's OEIS Frontend

A384592 a(n) = neg(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 A384592 #4 Jul 03 2025 16:21:11
%S A384592 0,-1,-6,-150,-5370,-800640,-190224111,-176905217601,-291774510700416,
%T A384592 -1732081174959242854,-20169113574187183899414,
%U A384592 -780617739418078727245260288,-64339950490716309261029142530799,-16501635484912814610105139633118406681
%N A384592 a(n) = neg(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 A384592 a(n) = (1/2)*((-1)^n A123745(n) - A384591(n)).
%e A384592 The rows of M(4) are (1,1,2,3), (3,1,1,2), (23,1,1), (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 A384592 z = 14;
%t A384592 v[n_] := Table[Fibonacci[k], {k, 1, n}];
%t A384592 u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
%t A384592 Table[Permanent[u[n]], {n, 1, z}]   (* A384591  *)
%t A384592 d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123745, with alternating signs *)
%t A384592 neg = (d - p)/2     (* A384592 *)
%t A384592 pos = (d + p)/2     (* A384593 *)
%Y A384592 Cf. A000045, A380661, A123745 (determinant), A384591 (permanent), A384593.
%K A384592 sign
%O A384592 1,3
%A A384592 _Clark Kimberling_, Jul 02 2025