A139037 Triangular sequence of coefficients of T_n(2^n*x) where T_n is the n-th Chebyshev polynomial.

1, 0, 2, -1, 0, 32, 0, -24, 0, 2048, 1, 0, -2048, 0, 524288, 0, 160, 0, -655360, 0, 536870912, -1, 0, 73728, 0, -805306368, 0, 2199023255552, 0, -896, 0, 117440512, 0, -3848290697216, 0, 36028797018963968, 1, 0, -2097152, 0, 687194767360, 0, -72057594037927936, 0, 2361183241434822606848, 0, 4608, 0

Offset

1,3

Comments

Old name was: A triangular sequence of coefficients Chebyshev T(x,n) as Besicovitch-Ursell polynomials (fractal orthogonal functions): f(x,n) = ChebyshevT[n,2^n*x]/2^(s*n).

Row sums are: 1, 2, 31, 2024, 522241, 536215712, 2198218022911, 36024948845706368, 2361111184527977349121, 618964706995596541734949376, 649035559893095618486323487178751, ... (cf A197320)

The integration of these functions is alternating orthogonal as are a lot of secondary Chebyshev polynomial sets: TableForm[Table[Integrate[(ChebyshevT[n, 2^n*x]/2^(s*n))*(ChebyshevT[m, 2^m*x]/2^(s*m))/Sqrt[1 -x^2], {x, -1, 1}], {n, 0, 10}, {m, 0, 10}]].

The fractal dimensional scaling here acts as a constant during integration.

These "biscuit" type functions are closely related to Weierstrass fractal and are usually constructed with a unit square "cartoon".

Graphing the polynomials shows that it is the "even" functions that are orthogonal. Orthogonal fractal systems are found in quantum fractals.

References

G. A. Edgar, Measure, Topology and Fractal Geometry, Springer-Verlag, New York, 1990, 202-206.

Links

Table of n, a(n) for n=1..48.

Daniel Wójcik, Iwo Białynicki-Birula, and Karol Zyczkowski, Time Evolution of Quantum Fractals, Phys. Rev. Lett. 85, 5022 - 5025 (2000).

Formula

s=Log[2]/Log[3]; f(x,n)=ChebyshevT[n,2^n*x]/2^(s*n); out_n,m=2^(s*n)*Coefficients(f(x,n))

Example

{1},
{0, 2},
{-1, 0, 32},
{0, -24, 0, 2048},
{1, 0, -2048,0, 524288},
{0, 160, 0, -655360, 0, 536870912},
{-1, 0, 73728, 0, -805306368, 0, 2199023255552},
{0, -896, 0, 117440512, 0, -3848290697216, 0, 36028797018963968},
{1, 0, -2097152, 0, 687194767360, 0, -72057594037927936, 0, 2361183241434822606848},
{0, 4608, 0, -16106127360, 0, 15199648742375424, 0, -5312662293228350865408, 0, 618970019642690137449562112},
{-1, 0, 52428800, 0, -439804651110400, 0,1291272085159668613120, 0, -1547425049106725343623905280,0, 649037107316853453566312041152512}

Mathematica

(* The polynomials: *) s = Log[2]/Log[3]; g = Table[ChebyshevT[n, 2^n*x]/2^(s*n), {n, 0, 10}];
(* the data: *) a = Table[CoefficientList[ChebyshevT[n, 2^n*x], x], {n, 0, 10}]; Flatten[a]
(* row sums: *) Table[Apply[Plus, CoefficientList[ChebyshevT[n, 2^n*x], x]], {n, 0, 10}];

Crossrefs

Sequence in context: A249570 A051652 A077019 * A108511 A261160 A270668

Adjacent sequences: A139034 A139035 A139036 * A139038 A139039 A139040

Keyword

tabl,sign

Author

Roger L. Bagula and Gary W. Adamson, May 31 2008

Extensions

Edited by Michel Marcus, May 30 2013

Status

approved