A359559 a(n) is the determinant 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, -3, -16, -36, -40, 20, 184, 400, 432, -112, -1472, -3136, -3328, 576, 9856, 20736, 21760, -2816, -59392, -123904, -129024, 13312, 333824, 692224, 716800, -61440, -1785856, -3686400, -3801088, 278528, 9207808, 18939904, 19464192, -1245184, -46137344, -94633984

Offset: 0

Author

Stefano Spezia, Jan 06 2023

Keywords

• sign

Examples

			a(3) = -16:
  [   1,  2*i, 3*i;
   -2*i,    1, 2*i;
   -3*i, -2*i,   1 ]
		

Links

• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A001792 (symmetric Toeplitz matrix), A143182.

Cf. A359560 (permanent), A359561, A359562.

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

Programs

Mathematica

Join[{1},Table[Det[ToeplitzMatrix[Join[{1},I Range[2,n]]]],{n,36}]]

PARI

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

Python

from sympy import Matrix, I
def A359559(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)]).det()*(1,-I,-1,I)[n&3] # Chai Wah Wu, Jan 25 2023

Formula

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

Conjectured formulas: (Start)

O.g.f.: (1 - 5*x + 9*x^2 - 12*x^3 + 10*x^4 - 4*x^5)/(1 - 2*x + 2*x^2)^3.

a(n) = 6*a(n-1) - 18*a(n-2) + 32*a(n-3) - 36*a(n-4) + 24*a(n-5) - 8*a(n-6) for n > 5.

E.g.f.: (2 + exp(x)*((1 + x)*(2 + x)*cos(x) - (1 + x + x^2)*sin(x)))/4. (End)