%I #17 Jul 07 2024 05:45:35

%S 1,0,-1,4,-44,-946,-8281,-592100,-25369920,-511563816,-55400732937

%N a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.

%e a(5) = -946:

%e [0, 1, 4, 2, 3]

%e [1, 0, 1, 4, 2]

%e [4, 1, 0, 1, 4]

%e [2, 4, 1, 0, 1]

%e [3, 2, 4, 1, 0]

%t a[0]=1; a[n_]:=Min[Table[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Range[n-1]],i]]]],{i,(n-1)!}]]; Array[a,11,0]

%Y Cf. A085750, A358323.

%Y Cf. A085807 (minimal permanent), A374280 (maximal), A374281 (maximal absolute value), A374282 (minimal nonzero absolute value), A374283 (maximal permanent).

%K sign,hard,more

%O 0,4

%A _Stefano Spezia_, Jul 02 2024