A137561 A triangular sequence of coefficients of the fixed point Chebyshev polynomials: p(x,n)=T(x,n)-x:A053120[x,n]-x.

1, -1, 0, -1, -1, 2, 0, -4, 0, 4, 1, -1, -8, 0, 8, 0, 4, 0, -20, 0, 16, -1, -1, 18, 0, -48, 0, 32, 0, -8, 0, 56, 0, -112, 0, 64, 1, -1, -32, 0, 160, 0, -256, 0, 128, 0, 8, 0, -120, 0, 432, 0, -576, 0, 256, -1, -1, 50, 0, -400, 0, 1120, 0, -1280, 0, 512

Offset

1,6

Comments

The row sums are all zero.

The idea of roots of polynomials of the sort came from the realization that in Umbral calculus for the expansion function:

p(x,t)=Sum(P(xd,n)*t^n/n!,{n,0,Infinity}];

to actually work there has to be a convergent limit:

Limit[P(x,n)*t^n/n!,n->Infinity]=0;

The idea that a point gets "trapped" in complex dynamics is the iterative:

Pc[x,n]=x

So if we look at polynomials as iterative steps, at a fixed point

the roots would be important dynamically.

References

Lennart Carleson, Theodore W. Gamelin, Complex Dynamics, Springer,New York,1993,Chapter II, page 27 ff

Links

Table of n, a(n) for n=1..66.

Formula

p(x,n)=T(x,n)-x:A053120[x,n]-x; out_n,m=Coefficients(A053120[x,n]-x).

Example

{1, -1},
{0},
{-1, -1, 2},
{0, -4, 0, 4},
{1, -1, -8, 0, 8},
{0, 4, 0, -20, 0, 16},
{-1, -1,18, 0, -48, 0, 32},
{0, -8, 0, 56, 0, -112, 0, 64},
{1, -1, -32, 0, 160, 0, -256, 0, 128},
{0, 8, 0, -120, 0, 432, 0, -576, 0, 256},
{-1, -1,50, 0, -400, 0, 1120, 0, -1280, 0, 512}

Mathematica

Table[ChebyshevT[n, x] - x, {n, 0, 10}]; a = Table[CoefficientList[ChebyshevT[n, x] - x, x], {n, 0, 10}]; Flatten[{{1, -1}, {0}, {-1, -1, 2}, {0, -4, 0, 4}, {1, -1, -8, 0, 8}, {0, 4, 0, -20, 0, 16}, {-1, -1, 18, 0, -48, 0, 32}, {0, -8, 0, 56, 0, -112, 0, 64}, {1, -1, -32, 0, 160, 0, -256, 0, 128}, {0, 8, 0, -120, 0, 432, 0, -576, 0, 256}, {-1, -1, 50, 0, -400, 0, 1120, 0, -1280, 0, 512}}]

Crossrefs

Sequence in context: A364105 A046769 A145893 * A337605 A082024 A337599

Adjacent sequences: A137558 A137559 A137560 * A137562 A137563 A137564

Keyword

tabl,uned,sign

Author

Roger L. Bagula, Apr 25 2008

Status

approved