A384075 -id:A384075 - cp's OEIS frontend

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.

1, 1, 153, 2668, 250200, 19423560, 2515242520, 404114856640, 84196030473216, 21703670967664000, 6808856052755927808, 2552126898198385479168, 1126590812208410998119424, 578462173661889165983466496, 341831898528862885226121600000

Offset: 1

Clark Kimberling, May 22 2025

nonn

			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.

Cf. A193678 (determinant), A384075 (permanent), A380661, A384077, A384078.

z = 19;
v[n_] := Table[2 k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A384074  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A193678,, with alternating signs *)
neg = (d - p)/2   (* A384075 *)
pos = (d + p)/2   (* A384076 *)

a(n) = (1/2)*(-(-1)^n*A193678(n) + A384074(n)).

A384074 a(n) = permanent of the n X n circulant matrix with (row 1) = (1, 3, 5, 7, ..., 2n - 1).

Original entry on oeis.org

1, 10, 198, 7384, 450400, 40340112, 4977778288, 810377196928, 168292881301248, 43412461935328000, 13617419946361149952, 5104272056570488986624, 2253180383840385394370560, 1156924438353338246938200064, 683663789883272270452243200000

Offset: 1

Clark Kimberling, May 22 2025

nonn

Cf. A005408, A193678 (determinant), A384075, A384076, A384077, A384078.

z = 16; v[n_] := Table[2 k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
Table[Simplify[Permanent[u[n]]], {n, 1, z}]

cp's OEIS Frontend

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.

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