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A384077 a(n) = neg(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

%I A384077 #18 Jun 18 2025 22:07:23
%S A384077 0,-9,-153,-2668,-200200,-20916552,-2515242520,-404114856640,
%T A384077 -84096850828032,-21708790967664000,-6808856052755927808,
%U A384077 -2552126898198385479168,-1126589571631974396251136,-578462264691449080954733568,-341831898528862885226121600000
%N A384077 a(n) = neg(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.
%F A384077 a(n) = (1/2)*(s(n)*A193678(n) - A384074(n)), where s(n) = (-1)^((2*n+(-1)^n-1)/4).
%e A384077 The rows of M(4) are (1,3,5,7), (3,5,7,1), (5,7,1,3), (7,1,3,5); determinant(M(4)) = 2048; permanent(M(4)) = 7384, so neg(M(4)) = (7384 - 2048)/2 = -2668 and pos(M(4)) = (7384+2048)/2 = 4716.
%t A384077 z = 15;
%t A384077 v[n_] := Table[2 k + 1, {k, 0, n - 1}];
%t A384077 u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
%t A384077 p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A384074 *)
%t A384077 d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A193678 up to signs *)
%t A384077 neg = (d - p)/2    (* A384077 *)
%t A384077 pos = (d + p)/2    (* A384078 *)
%Y A384077 Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384078.
%K A384077 sign
%O A384077 1,2
%A A384077 _Clark Kimberling_, May 29 2025