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A384075 a(n) = neg(M(n)), where M(n) is the n X n 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 A384075 #10 Jun 05 2025 00:48:48
%S A384075 0,-9,-45,-4716,-200200,-20916552,-2462535768,-406262340288,
%T A384075 -84096850828032,-21708790967664000,-6808563893605222144,
%U A384075 -2552145158372103507456,-1126589571631974396251136,-578462264691449080954733568,-341831891354409385226121600000
%N A384075 a(n) = neg(M(n)), where M(n) is the n X n 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 A384075 a(n) = (1/2)*((-1)^n*A193678(n) - A384074(n)).
%e A384075 The rows of M(4) are (1,3,5,7), (7,1,3,5), (5,7,1,3), (3,5,7,1); determinant(M(4)) = -4716; permanent(M(4)) = 2668, so neg(M(4)) = (-2048 - 7384)/2 = -4716 and pos(M(4)) = (-2048 + 7384)/2 = 2668.
%t A384075 z = 19;
%t A384075 v[n_] := Table[2 k + 1, {k, 0, n - 1}];
%t A384075 u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
%t A384075 p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A384074  *)
%t A384075 d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A193678, with alternating signs *)
%t A384075 neg = (d - p)/2   (* A384075 *)
%t A384075 pos = (d + p)/2   (* A384076 *)
%Y A384075 Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384077, A384078.
%K A384075 sign
%O A384075 1,2
%A A384075 _Clark Kimberling_, May 22 2025