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A123940 A Caratheodory Fejer Theorem set of matrices whose characteristic polynomials produce a triangular sequence: {{a[n],...,a[0]}, {a[n-1],..a[0],0}, ..., {a[0],0,...,0}}. 0
1, 1, -1, -1, -1, 1, -1, 0, 3, -1, 1, 1, -4, -4, 1, 1, 0, -6, 0, 8, -1, -1, -1, 7, 7, -12, -12, 1, -1, 0, 9, 0, -25, 0, 21, -1, 1, 1, -10, -10, 32, 32, -33, -33, 1, 1, 0, -12, 0, 51, 0, -90, 0, 55, -1, -1, -1, 13, 13, -61, -61, 122, 122, -88, -88, 1, -1, 0, 15, 0, -86, 0, 234, 0, -300, 0, 144, -1, 1, 1, -16, -16, 99, 99, -295, -295, 422, 422 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

As I was able to convert the Hankel matrix form into the Steinbach so I was able to get an Hankel-Fibonacci of this Caratheodory Fejer type matrix: 1 X 1 {{1}} 2 X 2 {1, 1}, {1, 0}}, 3 X 3 {{2, 1, 1}, {1, 1, 0}, {1, 0, 0}}, 4 X 4 {{3,2, 1, 1}, {2, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}}, 5 X 5 {{5, 3, 2, 1, 1}, {3, 2,1, 1, 0}, {2, 1, 1, 0, 0}, {1, 1, 0, 0, 0}, {1, 0, 0, 0, 0}}, 6 X 6 {{8, 5, 3, 2, 1, 1}, {5, 3, 2, 1, 1, 0}, {3, 2, 1, 1, 0, 0}, {2, 1,1, 0, 0, 0}, {1, 1, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0}}

REFERENCES

Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26

http://links.jstor.org/sici?sici=0036-1429(198204)19%3A2%3C358%3ARPCABT%3E2.0.CO%3 B2-K Real Polynomial Chebyshev Approximation by the Caratheodory-Fejer Method Martin H. Gutknecht, Lloyd N. Trefethen SIAM Journal on Numerical Analysis, Vol. 19, No. 2 (Apr., 1982), pp. 358-371

LINKS

Table of n, a(n) for n=1..88.

FORMULA

a(n,m)=Table[If[n +m - 1 > d, 0, Fibonacci[d - (n + m - 1) + 1]], {n, 1, d}, {m, 1, d}] a(i,j)->p(n,x) p(n,k)->t(n,m)

EXAMPLE

Triangular Sequence:

{1},

{1, -1},

{-1, -1, 1},

{-1, 0, 3, -1},

{1, 1, -4, -4, 1},

{1, 0, -6, 0, 8, -1},

{-1, -1, 7, 7, -12, -12,1},

{-1, 0, 9, 0, -25, 0, 21, -1},

{1, 1, -10, -10, 32, 32, -33, -33,1},

Polynomials:

1,

1 - x,

-1- x + x^2,

-1 + 3 x^2 - x^3,

1 + x - 4 x^2 - 4x^3 + x^4,

1 - 6 x^2 + 8 x^4 - x^5,

-1 - x + 7 x^2 + 7 x^3 - 12 x^4 - 12 x^5 + x^6

MATHEMATICA

An[d_] := Table[If[n + m - 1 > d, 0, Fibonacci[d - (n + m - 1) + 1]], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]

CROSSREFS

Sequence in context: A318873 A346403 A295924 * A339969 A204120 A342666

Adjacent sequences:  A123937 A123938 A123939 * A123941 A123942 A123943

KEYWORD

uned,tabl,sign

AUTHOR

Roger L. Bagula and Gary W. Adamson, Oct 25 2006

STATUS

approved

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Last modified July 29 07:28 EDT 2021. Contains 346340 sequences. (Running on oeis4.)