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 A137363 Triangular sequence of coefficients based on a Hilbert Transform of A053120: Chebyshev T(x,n); Coefficients(A053120[n,m])-Floor[Imaginary part of( HilbertTransform(A053120(n,m))];. 0
 1, 0, 1, -1, -1, 2, 4, -3, -3, 4, 1, 6, -8, -9, 8, 7, 5, 15, -20, -20, 16, -1, -3, 18, 37, -48, -46, 32, 26, -6, -19, 57, 95, -112, -99, 64, 1, 16, -32, -80, 160, 233, -256, -213, 128, 86, 9, 54, -120, -254, 432, 566, -576, -450, 256, -1, 14, 50, 174, -400, -746, 1120, 1344, -1280, -947, 512 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Row sums are: {1, 1, 0, 2, -2, 3, -11, 6, -43, 3, -160} This Hilbert transform/ operator has the property that to sign the a-b and a+b and the absolute value row sum for both is: ( called isobaric by Olver) {1, 1, 4, 14, 32, 83, 185, 478, 1119, 2803, 6588} REFERENCES Wilbur R. LePage, Complex Variables and the Laplace Transform for Engineers,Dover, New York,1961, page 225. P. J. Olver, Classical Invariant Theory, Cambridge Univ. Press, p. 222. http://jowett.home.cern.ch/jowett/Mathematica/Accelerator/Hilbert.nb LINKS FORMULA Coefficients(A053120[n,m])-Floor[Imaginary part of( HilbertTransform(A053120(n,m))]; EXAMPLE a-b: {1}, {0, 1}, {-1, -1, 2}, {4, -3, -3, 4}, {1, 6, -8, -9, 8}, {7, 5, 15, -20, -20, 16}, {-1, -3, 18, 37, -48, -46, 32}, {26, -6, -19, 57, 95, -112, -99, 64}, {1, 16, -32, -80,160, 233, -256, -213, 128}, {86, 9, 54, -120, -254, 432, 566, -576, -450,256}, {-1, 14, 50, 174, -400, -746, 1120, 1344, -1280, -947, 512} a+b: {1}, {0, 1}, {-1, 1, 2}, {-4, -3, 3, 4}, {1, -6, -8, 9, 8}, {-7, 5, -15, -20, 20, 16}, {-1, 3, 18, -37, -48, 46, 32}, {-26, -8, 19, 55, -95, -112,99, 64}, {1, -16, -32, 80, 160, -233, -256, 213, 128}, {-86, 9, -54, -120, 254, 432, -566, -576, 450, 256}, {-1, -14, 50, -174, -400,746, 1120, -1344, -1280, 947, 512} MATHEMATICA HilbertTransform[x_List] := Module[{nx, n, y}, nx = Length[x]; xn = If[EvenQ[nx], x, Append[x, 0]]; n = Length[xn]; y = Fourier[xn]; h = Flatten[{1, Table[2, {k, 2, n/2}], 1, Table[0, {k, n/2 + 2, n}]}]; Take[InverseFourier[h y], nx]]; a = Table[CoefficientList[ChebyshevT[n, x], x], {n, 0, 10}]; b = Table[Floor[Im[ HilbertTransform[CoefficientList[ChebyshevT[n, x], x]]]], {n, 0, 10}]; a-b CROSSREFS Cf. A053120. Sequence in context: A308182 A220080 A299920 * A110549 A174574 A161413 Adjacent sequences:  A137360 A137361 A137362 * A137364 A137365 A137366 KEYWORD tabl,uned,sign AUTHOR Roger L. Bagula, Apr 26 2008 STATUS approved

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Last modified November 24 10:24 EST 2020. Contains 338612 sequences. (Running on oeis4.)