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 A137336 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}]. 0
 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; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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}] REFERENCES Rosenblum and Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18-19 LINKS FORMULA Expansion of (-2*x*t+t^1)/(1-2*xt+t^2) EXAMPLE {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} MATHEMATICA 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}] CROSSREFS Sequence in context: A143424 A130125 A214809 * A115322 A053117 A121448 Adjacent sequences:  A137333 A137334 A137335 * A137337 A137338 A137339 KEYWORD uned,tabl,sign AUTHOR Roger L. Bagula, Apr 07 2008 STATUS approved

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