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Search: id:A137336
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| A137336 |
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Triangular sequence of coefficients from expansion of the U(x,n) one's opposite expansion: 1/(1-2*xt+t^2)=Sum[U(x,n),{n,0,Infinity}]; 1/(1-2*xt+t^2)+(-2*x*t+t^1)/(1-2*xt+t^2)=1 so that: (-2*x*t+t^1)/(1-2*xt+t^2)=Sum[p(x,n),{n,0,Infinity}]. |
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1
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| 0, 0, -2, 1, 0, -4, 0, 4, 0, -8, -1, 0, 12, 0, -16, 0, -6, 0, 32, 0, -32, 1, 0, -24, 0, 80, 0, -64, 0, 8, 0, -80, 0, 192, 0, -128, -1, 0, 40, 0, -240, 0, 448, 0, -256, 0, -10, 0, 160, 0, -672, 0, 1024, 0, -512, 1, 0, -60, 0, 560, 0, -1792, 0, 2304, 0, -1024
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row sums are:
{0, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11};
This polynomial set appears to be a Chebyshev related set of polynomials.
This sequence was suggested by the fact that except for zero the expansion of
f(x,t)=(-x*t+t^2)/(1-2*x*t+t^2) =Sum[q(x,n),{n,0,Infinity}]
is
q(x,n)= -T(x,n).
Integration of the recursive polynomials shows alternating orthogonality:
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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REFERENCES
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Rosenblum and Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18-19
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FORMULA
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Expansion of (-2*x*t+t^1)/(1-2*xt+t^2)
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EXAMPLE
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{0},
{0, -2},
{1, 0, -4},
{0, 4, 0, -8},
{-1, 0, 12, 0, -16},
{0, -6, 0, 32,0, -32},
{1, 0, -24, 0, 80, 0, -64},
{0, 8, 0, -80, 0, 192,0, -128},
{-1, 0, 40, 0, -240, 0, 448, 0, -256},
{0, -10, 0, 160, 0, -672, 0, 1024, 0, -512},
{1, 0, -60, 0, 560, 0, -1792, 0, 2304, 0, -1024}
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MATHEMATICA
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Clear[p] p[t_] = (-2*x*t + t^2)/(1 - 2*x*t + t^2); Table[ ExpandAll[SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Join[{{0}}, Table[ CoefficientList[ExpandAll[SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], x], {n, 0, 10}]]; Flatten[a] (* polynomial recursion: needs first three terms*) Clear[p] p[x, 0] = 0; p[x, 1] = -2*x; p[x, 2] = 1 - 4*x^2; p[x_, n_] := p[x, n] = 2*x*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, Length[g] - 1}]
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CROSSREFS
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Adjacent sequences: A137333 A137334 A137335 this_sequence A137337 A137338 A137339
Sequence in context: A088850 A143424 A130125 this_sequence A115322 A053117 A121448
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 07 2008
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