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

%I A137561 #3 Mar 30 2012 17:34:26
%S A137561 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,
%T A137561 32,0,-8,0,56,0,-112,0,64,1,-1,-32,0,160,0,-256,0,128,0,8,0,-120,0,
%U A137561 432,0,-576,0,256,-1,-1,50,0,-400,0,1120,0,-1280,0,512
%N A137561 A triangular sequence of coefficients of the fixed point Chebyshev polynomials: p(x,n)=T(x,n)-x:A053120[x,n]-x.
%C A137561 The row sums are all zero.
%C A137561 The idea of roots of polynomials of the sort came from the realization that in Umbral calculus for the expansion function:
%C A137561 p(x,t)=Sum(P(xd,n)*t^n/n!,{n,0,Infinity}];
%C A137561 to actually work there has to be a convergent limit:
%C A137561 Limit[P(x,n)*t^n/n!,n->Infinity]=0;
%C A137561 The idea that a point gets "trapped" in complex dynamics is the iterative:
%C A137561 Pc[x,n]=x
%C A137561 So if we look at polynomials as iterative steps, at a fixed point
%C A137561 the roots would be important dynamically.
%D A137561 Lennart Carleson, Theodore W. Gamelin, Complex Dynamics, Springer,New York,1993,Chapter II, page 27 ff
%F A137561 p(x,n)=T(x,n)-x:A053120[x,n]-x; out_n,m=Coefficients(A053120[x,n]-x).
%e A137561 {1, -1},
%e A137561 {0},
%e A137561 {-1, -1, 2},
%e A137561 {0, -4, 0, 4},
%e A137561 {1, -1, -8, 0, 8},
%e A137561 {0, 4, 0, -20, 0, 16},
%e A137561 {-1, -1,18, 0, -48, 0, 32},
%e A137561 {0, -8, 0, 56, 0, -112, 0, 64},
%e A137561 {1, -1, -32, 0, 160, 0, -256, 0, 128},
%e A137561 {0, 8, 0, -120, 0, 432, 0, -576, 0, 256},
%e A137561 {-1, -1,50, 0, -400, 0, 1120, 0, -1280, 0, 512}
%t A137561 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}}]
%K A137561 tabl,uned,sign
%O A137561 1,6
%A A137561 _Roger L. Bagula_, Apr 25 2008