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}}.

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

Offset

1,9

References

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

Links

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

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

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