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A140883
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A coefficient triangular sequence made from the ChebyshevT polynomials, T(x,n) and their toral inverse (or reversed coefficient) polynomial x^n*T(1/x,n): p(x,n)=T[x,n]+x^n*T(1/x,n).
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0
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2, 1, 1, 1, 0, 1, 4, -3, -3, 4, 9, 0, -16, 0, 9, 16, 5, -20, -20, 5, 16, 31, 0, -30, 0, -30, 0, 31, 64, -7, -112, 56, 56, -112, -7, 64, 129, 0, -288, 0, 320, 0, -288, 0, 129, 256, 9, -576, -120, 432, 432, -120, -576, 9, 256, 511, 0, -1230, 0, 720, 0, 720, 0, -1230, 0, 511
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Row sums are all two.
All this does is make a symmetrical coefficient triangle
since the double integration is no where zero, they aren't orthogonal;
Table[Integrate[p[x, n]*p[x, m]/
Sqrt[1 - x^2], {x, -1, 1}], {n, 0, 10}, {m, 0, 10}]
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FORMULA
| p(x,n)=T[x,n]+x^n*T(1/x,n); Out_n,m=Coefficients(p(x,n)).
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EXAMPLE
| {2},
{1, 1},
{1, 0, 1},
{4, -3, -3, 4},
{9, 0, -16, 0, 9},
{16, 5, -20, -20, 5, 16},
{31, 0, -30, 0, -30, 0, 31},
{64, -7, -112, 56, 56, -112, -7, 64},
{129, 0, -288, 0, 320, 0, -288, 0, 129},
{256, 9, -576, -120, 432, 432, -120, -576, 9, 256},
{511, 0, -1230, 0, 720, 0, 720, 0, -1230, 0, 511}
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MATHEMATICA
| Clear[p, x, n, m, a]; p[x_, n_] := ChebyshevT[n, x] + ExpandAll[x^n*ChebyshevT[n, 1/x]]; Table[p[x, n], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
| Cf. A053120.
Sequence in context: A068029 A158208 A117274 * A064744 A135997 A026609
Adjacent sequences: A140880 A140881 A140882 * A140884 A140885 A140886
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jul 22 2008
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