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old recursive polynomial set into matrices


Here is a classical Chebyshev polynomial recursion as triangular sequence
made into a matrix set and the result
reconverted into a triangular sequence that "checks:".


Clear[p, a, b, An]
p[-1, x] = 0; p[0, x] = 1; p[1, x] = x + 1;
p[k_, x_] := p[k, x] = 2*x*p[k - 1, x] - p[k - 2, x]
Table[Expand[p[n, x]], {n, 0, 10}]
Table[Sum[
      CoefficientList[p[n, x], x][[m]], {m, 1, Length[CoefficientList[
        p[n, x], x]]}], {n, 0, 15}]
w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]
Flatten[w]
An[d_] := Table[If[n == d && m < n, -w[[n]][[d - m + 1]], If[m == n + 1, 1,
     0]], {n, 1, d}, {m, 1, d}]
Table[An[d], {d, 1, 10}]
Table[Det[An[d]]/2^(d - 2), {d, 1, 10}]
Table[ExpandAll[y^(d - 1)*(CharacteristicPolynomial[An[d], x] /. x -> 
1/y)] \
/. 1/y -> 1, {d, 1, 10}]
Table[NSolve[CharacteristicPolynomial[An[d], x] == 0, x], {d, 2, 10}]
Table[Abs[Flatten[b][[n]]] - Abs[Flatten[w][[
      n]]], {n, 1, Min[Length[Flatten[b]], Length[Flatten[w]]]}]


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