A359560 a(n) is the permanent of an n X n Hermitian Toeplitz matrix whose first row consists of 1, 2*i, ..., n*i, where i denotes the imaginary unit.

1, 1, 5, 18, 360, 2800, 151424, 1926704, 218991568, 3961998320, 815094714320, 19339258670304, 6524060415099520, 192715406460607360, 99364368150722162944, 3525158026102570745600, 2635328330670632415828224, 109381927750670379873854720, 113797518402277434839782802688

Offset: 0

Author

Stefano Spezia, Jan 06 2023

Keywords

• nonn

Examples

			a(3) = 18:
  [   1,  2*i, 3*i;
   -2*i,    1, 2*i;
   -3*i, -2*i,   1 ]
		

Links

• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A143182, A204235 (symmetric Toeplitz matrix).

Cf. A359559 (determinant), A359561, A359562.

Cf. A359614 (minimal), A359615 (maximal).

Programs

Maple

A359560 := proc(n)
    local T,c,r ;
    if n =0 then
        return 1 ;
    end if;
    T := Matrix(n,n,shape=hermitian) ;
    T[1,1] := 1 ;
    for c from 2 to n do
        T[1,c] := c*I ;
    end do:
    for r from 2 to n do
        for c from r to n do
            T[r,c] := T[r-1,c-1] ;
        end do:
    end do:
    LinearAlgebra[Permanent](T) ;
    simplify(%) ;
end proc:
seq(A359560(n),n=0..15) ; # R. J. Mathar, Jan 31 2023

Mathematica

Join[{1},Table[Permanent[ToeplitzMatrix[Join[{1},I Range[2,n]]]],{n,18}]]

PARI

a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 1, if (iMichel Marcus, Jan 20 2023

Python

from sympy import Matrix, I
def A359560(n): return Matrix(n,n,[i-j+(1 if i>j else -1) if i!=j else I for i in range(n) for j in range(n)]).per()*(1,-I,-1,I)[n&3] if n else 1 # Chai Wah Wu, Jan 25 2023

Formula

A359614(n) <= a(n) <= A359615(n).