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A053120 Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order). 176
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, -7, 0, 56, 0, -112, 0, 64, 1, 0, -32, 0, 160, 0, -256, 0, 128, 0, 9, 0, -120, 0, 432, 0, -576, 0, 256, -1, 0, 50, 0, -400, 0, 1120, 0, -1280, 0, 512, 0, -11, 0, 220, 0, -1232, 0, 2816, 0, -2816 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

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).

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

From Wolfdieter Lang, Oct 21 2013: (Start)

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.

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)

REFERENCES

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.

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

LINKS

T. D. Noe, Rows 0 to 100 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p.795.

P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620, 2014.- From Tom Copeland, Oct 11 2014

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Wolfdieter Lang, Rows n=0..20.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

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.

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).

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).

EXAMPLE

The triangle a(n,m) begins:

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

0:   1

1:   0  1

2:  -1  0   2

3:   0 -3   0    4

4:   1  0  -8    0     8

5:   0  5   0  -20     0   16

6:  -1  0  18    0   -48    0    32

7:   0 -7   0   56     0 -112     0    64

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

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

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

... Reformatted and extended - Wolfdieter Lang, Oct 21 2013.

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

MAPLE

with(orthopoly) ;

A053120 := proc(n, k)

    T(n, x) ;

    coeftayl(%, x=0, k) ;

end proc: # R. J. Mathar, Jun 30 2013

MATHEMATICA

t[n_, k_] := Coefficient[ ChebyshevT[n, x], x, k]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Jean-Fran├žois Alcover, Jan 16 2012 *)

PROG

(MAGMA) &cat[ Coefficients(ChebyshevT(n)): n in [0..11] ]; // Klaus Brockhaus, Mar 08 2008

(PARI) for(n=0, 5, P=polchebyshev(n); for(k=0, n, print1(polcoeff(P, k)", "))) \\ Charles R Greathouse IV, Jan 16 2012

CROSSREFS

Cf. A039991, A000012, A001333.

Sequence in context: A223707 A046767 A115720 * A008743 A029179 A008721

Adjacent sequences:  A053117 A053118 A053119 * A053121 A053122 A053123

KEYWORD

sign,tabl,nice,easy,changed

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified October 25 07:15 EDT 2014. Contains 248518 sequences.