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

0,3

Links

Table of n, a(n) for n=0..18.

Wikipedia, Toeplitz Matrix

Formula

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

Example

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

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}]]

Prog

(PARI) a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 1, if (i<j, I*(j-i+1), I*(j-i-1))))); \\ Michel 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

Crossrefs

Cf. A143182, A204235 (symmetric Toeplitz matrix).

Cf. A359559 (determinant), A359561, A359562.

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

Sequence in context: A352663 A203180 A359616 * A139243 A347670 A139237

Adjacent sequences: A359557 A359558 A359559 * A359561 A359562 A359563

Keyword

nonn

Author

Stefano Spezia, Jan 06 2023

Status

approved