cp's OEIS Frontend

A384596 a(n) = pos(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 pos(M(n)) is the positive part of the determinant of M(n); see A380661.

%I A384596 #4 Jul 15 2025 00:17:31
%S A384596 1,1,36,630,75264,10309104,6689940744,6609305472651,25797682556382400,
%T A384596 181805125100075828614,4497447436889592767655228,
%U A384596 225317753810449180044272832000,36410024238337826166260377355303568,12904889422278677354475665889049659231531
%N A384596 a(n) = pos(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 pos(M(n)) is the positive part of the determinant of M(n); see A380661.
%F A384596 a(n) = (1/2)*(A118704(n) + A384594(n)).
%e A384596 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 A384596 z = 14;
%t A384596 v[n_] := Table[Fibonacci[k], {k, 2, n + 1}];
%t A384596 u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
%t A384596 p = Table[Permanent[u[n]], {n, 1, z}]   (* A384594 *)
%t A384596 d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A118704 *)
%t A384596 neg = (d - p)/2     (* A384595 *)
%t A384596 pos = (d + p)/2     (* A384596 *)
%Y A384596 Cf. A000045, A118704 (determinant), A384594 (permanent), A384595.
%K A384596 nonn
%O A384596 1,3
%A A384596 _Clark Kimberling_, Jul 10 2025