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A384078 a(n) = pos(M(n)), where M(n) is the n X n left 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 A384078 #15 Jun 18 2025 22:09:51
%S A384078 1,1,45,4716,250200,19423560,2462535768,406262340288,84196030473216,
%T A384078 21703670967664000,6808563893605222144,2552145158372103507456,
%U A384078 1126590812208410998119424,578462173661889165983466496,341831891354409385226121600000
%N A384078 a(n) = pos(M(n)), where M(n) is the n X n left 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 A384078 a(n) = (1/2)*(s(n)*A193678(n) - A384074(n)), where s(n) = (-1)^((2*n+(-1)^n-1)/4).
%e A384078 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 A384078 z = 15;
%t A384078 v[n_] := Table[2 k + 1, {k, 0, n - 1}];
%t A384078 u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
%t A384078 p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]     (* A384074 *)
%t A384078 d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A193678 up to signs *)
%t A384078 neg = (d - p)/2    (* A384077 *)
%t A384078 pos = (d + p)/2    (* A384078 *)
%Y A384078 Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384077.
%K A384078 nonn
%O A384078 1,3
%A A384078 _Clark Kimberling_, Jun 01 2025