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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))];.

%I A137363 #3 Mar 30 2012 17:34:26
%S A137363 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,
%T A137363 -48,-46,32,26,-6,-19,57,95,-112,-99,64,1,16,-32,-80,160,233,-256,
%U A137363 -213,128,86,9,54,-120,-254,432,566,-576,-450,256,-1,14,50,174,-400,-746,1120,1344,-1280,-947,512
%N 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))];.
%C A137363 Row sums are:
%C A137363 {1, 1, 0, 2, -2, 3, -11, 6, -43, 3, -160}
%C A137363 This Hilbert transform/ operator has the property
%C A137363 that to sign the a-b and a+b and the absolute value row sum for both is: ( called isobaric by Olver)
%C A137363 {1, 1, 4, 14, 32, 83, 185, 478, 1119, 2803, 6588}
%D A137363 Wilbur R. LePage, Complex Variables and the Laplace Transform for Engineers,Dover, New York,1961, page 225.
%D A137363 P. J. Olver, Classical Invariant Theory, Cambridge Univ. Press, p. 222.
%D A137363 http://jowett.home.cern.ch/jowett/Mathematica/Accelerator/Hilbert.nb
%F A137363 Coefficients(A053120[n,m])-Floor[Imaginary part of( HilbertTransform(A053120(n,m))];
%e A137363 a-b:
%e A137363 {1},
%e A137363 {0, 1},
%e A137363 {-1, -1, 2},
%e A137363 {4, -3, -3, 4},
%e A137363 {1, 6, -8, -9, 8},
%e A137363 {7, 5, 15, -20, -20, 16},
%e A137363 {-1, -3, 18, 37, -48, -46, 32},
%e A137363 {26, -6, -19, 57, 95, -112, -99, 64},
%e A137363 {1, 16, -32, -80,160, 233, -256, -213, 128},
%e A137363 {86, 9, 54, -120, -254, 432, 566, -576, -450,256},
%e A137363 {-1, 14, 50, 174, -400, -746, 1120, 1344, -1280, -947, 512}
%e A137363 a+b:
%e A137363 {1},
%e A137363 {0, 1},
%e A137363 {-1, 1, 2},
%e A137363 {-4, -3, 3, 4},
%e A137363 {1, -6, -8, 9, 8},
%e A137363 {-7, 5, -15, -20, 20, 16},
%e A137363 {-1, 3, 18, -37, -48, 46, 32},
%e A137363 {-26, -8, 19, 55, -95, -112,99, 64},
%e A137363 {1, -16, -32, 80, 160, -233, -256, 213, 128},
%e A137363 {-86, 9, -54, -120, 254, 432, -566, -576, 450, 256},
%e A137363 {-1, -14, 50, -174, -400,746, 1120, -1344, -1280, 947, 512}
%t A137363 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
%Y A137363 Cf. A053120.
%K A137363 tabl,uned,sign
%O A137363 1,6
%A A137363 _Roger L. Bagula_, Apr 26 2008