login
A137561
A triangular sequence of coefficients of the fixed point Chebyshev polynomials: p(x,n)=T(x,n)-x:A053120[x,n]-x.
0
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
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
KEYWORD
tabl,uned,sign
AUTHOR
Roger L. Bagula, Apr 25 2008
STATUS
approved