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%I #23 Oct 15 2020 03:14:17
%S 1,0,2,-1,0,32,0,-24,0,2048,1,0,-2048,0,524288,0,160,0,-655360,0,
%T 536870912,-1,0,73728,0,-805306368,0,2199023255552,0,-896,0,117440512,
%U 0,-3848290697216,0,36028797018963968,1,0,-2097152,0,687194767360,0,-72057594037927936,0,2361183241434822606848,0,4608,0
%N Triangular sequence of coefficients of T_n(2^n*x) where T_n is the n-th Chebyshev polynomial.
%C 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).
%C Row sums are: 1, 2, 31, 2024, 522241, 536215712, 2198218022911, 36024948845706368, 2361111184527977349121, 618964706995596541734949376, 649035559893095618486323487178751, ... (cf A197320)
%C 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}]].
%C The fractal dimensional scaling here acts as a constant during integration.
%C These "biscuit" type functions are closely related to Weierstrass fractal and are usually constructed with a unit square "cartoon".
%C Graphing the polynomials shows that it is the "even" functions that are orthogonal. Orthogonal fractal systems are found in quantum fractals.
%D G. A. Edgar, Measure, Topology and Fractal Geometry, Springer-Verlag, New York, 1990, 202-206.
%H Daniel Wójcik, Iwo Białynicki-Birula, and Karol Zyczkowski, <a href="http://www.cns.gatech.edu/~danek/preprints/prl.85.5022.pdf">Time Evolution of Quantum Fractals</a>, Phys. Rev. Lett. 85, 5022 - 5025 (2000).
%F 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))
%e {1},
%e {0, 2},
%e {-1, 0, 32},
%e {0, -24, 0, 2048},
%e {1, 0, -2048,0, 524288},
%e {0, 160, 0, -655360, 0, 536870912},
%e {-1, 0, 73728, 0, -805306368, 0, 2199023255552},
%e {0, -896, 0, 117440512, 0, -3848290697216, 0, 36028797018963968},
%e {1, 0, -2097152, 0, 687194767360, 0, -72057594037927936, 0, 2361183241434822606848},
%e {0, 4608, 0, -16106127360, 0, 15199648742375424, 0, -5312662293228350865408, 0, 618970019642690137449562112},
%e {-1, 0, 52428800, 0, -439804651110400, 0,1291272085159668613120, 0, -1547425049106725343623905280,0, 649037107316853453566312041152512}
%t (* The polynomials: *) s = Log[2]/Log[3]; g = Table[ChebyshevT[n, 2^n*x]/2^(s*n), {n, 0, 10}];
%t (* the data: *) a = Table[CoefficientList[ChebyshevT[n, 2^n*x], x], {n, 0, 10}]; Flatten[a]
%t (* row sums: *) Table[Apply[Plus, CoefficientList[ChebyshevT[n, 2^n*x], x]], {n, 0, 10}];
%K tabl,sign
%O 1,3
%A _Roger L. Bagula_ and _Gary W. Adamson_, May 31 2008
%E Edited by _Michel Marcus_, May 30 2013
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