login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

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: 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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