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A383775 a(n) = pos(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1, 2, ... , n), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

%I A383775 #6 May 27 2025 22:13:59
%S A383775 1,1,18,610,17550,744906,47282844,3918873376,410384120220,
%T A383775 53323552728000,8417451284317614,1586200451151892608,
%U A383775 351735343178101060906,90667504180193792086144,26884188746929397888775000,9086147134545912835276742656,3472279409772212369077001352888
%N A383775 a(n) = pos(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1, 2, ... , n), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.
%e A383775 The rows of M(4) are (1, 2, 3, 4), (2, 3, 4, 1), (3, 4, 1, 2), (4, 1, 2, 3); determinant(M(4)) = 160; permanent(M(4)) = 1060, so neg(M(4)) = (160 - 1060)/2 = -450 and pos(M(4)) = (160 + 1060)/2 = 610.
%t A383775 z = 18;
%t A383775 v[n_] := Table[k + 1, {k, 0, n - 1}];
%t A383775 u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
%t A383775 p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A085719 *)
%t A383775 d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182, with altered signs *)
%t A383775 neg = (d - p)/2   (* A383774 *)
%t A383775 pos = (d + p)/2   (* A383775 *)
%Y A383775 Cf. A052182 (determinant), A085719 (permanent), A380661, A383772, A383773, A383774.
%K A383775 nonn
%O A383775 1,3
%A A383775 _Clark Kimberling_, May 22 2025