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A139037
Triangular sequence of coefficients of T_n(2^n*x) where T_n is the n-th Chebyshev polynomial.
0
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
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
KEYWORD
tabl,sign
AUTHOR
EXTENSIONS
Edited by Michel Marcus, May 30 2013
STATUS
approved