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 A053120 Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order). 188

%I

%S 1,0,1,-1,0,2,0,-3,0,4,1,0,-8,0,8,0,5,0,-20,0,16,-1,0,18,0,-48,0,32,0,

%T -7,0,56,0,-112,0,64,1,0,-32,0,160,0,-256,0,128,0,9,0,-120,0,432,0,

%U -576,0,256,-1,0,50,0,-400,0,1120,0,-1280,0,512,0,-11,0,220,0,-1232,0,2816,0,-2816

%N Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order).

%C a(n,m) = A039991(n,n-m).

%C G.f. for row polynomials T(n,x) (signed triangle): (1-x*z)/(1-2*x*z+z^2). If unsigned:(1-x*z)/(1-2*x*z-z^2).

%C Row sums (signed triangle): A000012 (powers of 1). Row sums (unsigned triangle): A001333(n).

%C From _Wolfdieter Lang_, Oct 21 2013: (Start)

%C The row polynomials T(n,x) equal (S(n,2*x) - S(n-2,2*x))/2, n >= 0, with the row polynomials S from A049310, with S(-1,x) = 0, and S(-2,x) = -1.

%C The zeros of T(n,x) are x(n,k) = cos((2*k+1)*Pi/(2*n)), k = 0, 1, ... ,n-1, n >= 1. (End)

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available), p. 795.

%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

%H T. D. Noe, <a href="/A053120/b053120.txt">Rows 0 to 100 of triangle, flattened</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p.795.

%H P. Barry, A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry5/barry96s.html">Meixner-Type Results for Riordan Arrays and Associated Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.9.4, section 5.

%H P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014.- From _Tom Copeland_, Oct 11 2014

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

%H Wolfdieter Lang, <a href="/A053120/a053120.pdf">Rows n=0..20.</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F a(n, m) := 0 if n<m or n+m odd; a(n, m)= (-1)^n/2 if m=0 (n even); else a(n, m)=((-1)^((n+m)/ 2+m))*(2^(m-1))*n*binomial((n+m)/2-1, m-1)/m.

%F Recursion for n >= 2: a(n, m) = 2*a(n-1, m-1)-a(n-2, m), a(n, m)=0 if n<m, a(n, -1) := 0, a(0, 0)=1=a(1, 1).

%F G.f. for m-th column (signed triangle): 1/(1+x^2) if m=0 else (2^(m-1))*(x^m)*(1-x^2)/(1+x^2)^(m+1).

%e The triangle a(n,m) begins:

%e n\m 0 1 2 3 4 5 6 7 8 9 10

%e 0: 1

%e 1: 0 1

%e 2: -1 0 2

%e 3: 0 -3 0 4

%e 4: 1 0 -8 0 8

%e 5: 0 5 0 -20 0 16

%e 6: -1 0 18 0 -48 0 32

%e 7: 0 -7 0 56 0 -112 0 64

%e 8: 1 0 -32 0 160 0 -256 0 128

%e 9: 0 9 0 -120 0 432 0 -576 0 256

%e 10: -1 0 50 0 -400 0 1120 0 -1280 0 512

%e ... Reformatted and extended - _Wolfdieter Lang_, Oct 21 2013.

%e E.g., the fourth row (n=3) corresponds to the polynomial T(3,x) = -3*x + 4*x^3.

%p with(orthopoly) ;

%p A053120 := proc(n,k)

%p T(n,x) ;

%p coeftayl(%,x=0,k) ;

%p end proc: # _R. J. Mathar_, Jun 30 2013

%t t[n_, k_] := Coefficient[ ChebyshevT[n, x], x, k]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* _Jean-François Alcover_, Jan 16 2012 *)

%o (MAGMA) &cat[ Coefficients(ChebyshevT(n)): n in [0..11] ]; // _Klaus Brockhaus_, Mar 08 2008

%o (PARI) for(n=0,5,P=polchebyshev(n);for(k=0,n,print1(polcoeff(P,k)", "))) \\ _Charles R Greathouse IV_, Jan 16 2012

%o (Julia)

%o using Nemo

%o function A053120Row(n)

%o R, x = PolynomialRing(ZZ, "x")

%o p = chebyshev_t(n, x)

%o [coeff(p, j) for j in 0:n] end

%o for n in 0:6 A053120Row(n) |> println end # _Peter Luschny_, Mar 13 2018

%Y Cf. A039991, A000012, A001333.

%K sign,tabl,nice,easy

%O 0,6

%A _Wolfdieter Lang_

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Last modified November 20 19:02 EST 2018. Contains 317413 sequences. (Running on oeis4.)