1,6

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}

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

Coefficients(A053120[n,m])-Floor[Imaginary part of( HilbertTransform(A053120(n,m))];

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}

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

Cf. A053120.

Adjacent sequences: A137360 A137361 A137362 this_sequence A137364 A137365 A137366

Sequence in context: A084936 A099066 A021415 this_sequence A110549 A069655 A004574

tabl,uned,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 26 2008