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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
REFERENCES
Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26.
LINKS
Martin H. Gutknecht and Lloyd N. Trefethen, Real Polynomial Chebyshev Approximation by the Caratheodory-Fejer Method, SIAM Journal on Numerical Analysis, Vol. 19, No. 2 (Apr., 1982), pp. 358-371.
EXAMPLE
Triangle begins:
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 - 4*x^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: A357669 A361012 A363903 * A350447 A339969 A204120
KEYWORD
uned,tabl,sign
AUTHOR
STATUS
approved

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Last modified May 23 18:59 EDT 2024. Contains 372765 sequences. (Running on oeis4.)