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A384076 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

%I A384076 #12 Jun 11 2025 00:34:15
%S A384076 1,1,153,2668,250200,19423560,2515242520,404114856640,84196030473216,
%T A384076 21703670967664000,6808856052755927808,2552126898198385479168,
%U A384076 1126590812208410998119424,578462173661889165983466496,341831898528862885226121600000
%N A384076 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.
%F A384076 a(n) = (1/2)*(-(-1)^n*A193678(n) + A384074(n)).
%e A384076 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 A384076 z = 19;
%t A384076 v[n_] := Table[2 k + 1, {k, 0, n - 1}];
%t A384076 u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
%t A384076 p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A384074  *)
%t A384076 d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A193678,, with alternating signs *)
%t A384076 neg = (d - p)/2   (* A384075 *)
%t A384076 pos = (d + p)/2   (* A384076 *)
%Y A384076 Cf. A193678 (determinant), A384075 (permanent), A380661, A384077, A384078.
%K A384076 nonn
%O A384076 1,3
%A A384076 _Clark Kimberling_, May 22 2025