A359561 a(n) is the determinant of an n X n Hermitian Toeplitz matrix whose first row consists of n, (n-1)*i, (n-2)*i, ..., 3*i, 2*i, i, where i denotes the imaginary unit.

1, 1, 3, 0, -256, -5000, -46656, 941192, 67108864, 2066242608, 24000000000, -1659995174464, -142657607172096, -5964309791355136, -76196618232397824, 11210083593750000000, 1180591620717411303424, 62286325600853591655680, 839390038939659468275712, -213252813410122222659258368

Offset: 0

Author

Stefano Spezia, Jan 06 2023

Keywords

• sign

Examples

			a(3) = 0:
  [   3,  2*i,   i;
   -2*i,    3, 2*i;
     -i, -2*i,   3 ]
		

Links

• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A307783 (symmetric Toeplitz matrix).

Cf. A359559, A359560, A359562 (permanent).

Cf. A359616 (minimal), A359617 (maximal).

Programs

Maple

A359561 := proc(n)
    local T,c,r ;
    if n =0 then
        return 1 ;
    end if;
    T := Matrix(n,n) ;
    T[1,1] := n ;
    for c from 2 to n do
        T[1,c] := (n-c+1)*I ;
    end do:
    for r from 2 to n do
        for c from 1 to r-1 do
            T[r,c] := -T[c,r] ;
        end do:
        for c from r to n do
            T[r,c] := T[r-1,c-1] ;
        end do:
    end do:
    LinearAlgebra[Determinant](T) ;
    simplify(%) ;
end proc:
seq(A359561(n),n=0..25) ; # R. J. Mathar, Jan 31 2023

Mathematica

Join[{1},Table[Det[ToeplitzMatrix[Join[{n},I Reverse[Range[n-1]]]]],{n,19}]]

Python

from sympy import Matrix, I
def A359561(n): return Matrix(n,n,[(n+j-i if i>j else j-i-n) if i!=j else n*I for i in range(n) for j in range(n)]).det()*(1,-I,-1,I)[n&3] # Chai Wah Wu, Jan 25 2023

Formula

A359616(n) <= a(n) <= A359617(n).