|
| |
|
|
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.
|
|
7
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
Table of n, a(n) for n=0..36.
Wikipedia, Toeplitz Matrix
|
|
|
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)
|
|
|
EXAMPLE
|
a(3) = -16:
[ 1, 2*i, 3*i;
-2*i, 1, 2*i;
-3*i, -2*i, 1 ]
|
|
|
MATHEMATICA
|
Join[{1}, Table[Det[ToeplitzMatrix[Join[{1}, I Range[2, n]]]], {n, 36}]]
|
|
|
PROG
|
(PARI) a(n) = matdet(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 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
|
|
|
CROSSREFS
|
Cf. A001792 (symmetric Toeplitz matrix), A143182.
Cf. A359560 (permanent), A359561, A359562.
Cf. A359614 (minimal), A359615 (maximal).
Sequence in context: A013199 A162419 A322191 * A076153 A031302 A217130
Adjacent sequences: A359556 A359557 A359558 * A359560 A359561 A359562
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
Stefano Spezia, Jan 06 2023
|
|
|
STATUS
|
approved
|
| |
|
|
