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))];.

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

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

Table of n, a(n) for n=1..66.

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: A343766 A299920 A352962 * A110549 A174574 A161413

Adjacent sequences: A137360 A137361 A137362 * A137364 A137365 A137366

Keyword

tabl,uned,sign

Author

Roger L. Bagula, Apr 26 2008

Status

approved