login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A053120 Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order). 204
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

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)

From Wolfdieter Lang, Jan 03 2020 and Paul Weisenhorn: (Start)

The (sub)diagonal sequences {D_{2*k}(m)}_{m >= 0}, for k >= 0, have o.g.f. GD_{2*k}(x) = (-1)^k*(1-x)/(1-2*x)^(k+1), for k >= 0, and  GD_{2*k+1}(x) = 0, for k >= 0. This follows from their o.g.f. GGD(z, x) := Sum_{k>=0}  GD_k(x)*z^n which is obtained from the o.g.f of the T-triangle GT(z, x) = (1-x*z)/(1 - 2*x + z^2) (see the formula section) by GGD(z, x) = GT(z, x/z).

The explicit form is then D_{2*k}(m) =  (-1)^k, for m = 0, and

  (-1)^k*(2*k+m)*2^(m-1)*risefac(k+1, m-1)/m!, for m >= 1, with the rising factorial risefac(x, n). (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.

F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 pp. 77, 105.

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

TableCurve 2D, Automated curve fitting and equation discovery, Version 5.01 for Windows, User's Manual, Chebyshev Series Polynomials and Rationals, pages 12-21 - 12-24, SYSTAT Software, Inc., Richmond, WA, 2002.

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. Barry, A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 5.

T. Copeland, Addendum to Elliptic Lie Triad

P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 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.

Wikipedia, Chebyshev polynomials

Wolfdieter Lang, Rows n = 0..20.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

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).

a(n, m) := 0 if n < m or n+m odd; a(n, m) = (-1)^(n/2) if m=0 (n even); otherwise 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, otherwise (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

(Julia)

using Nemo

function A053120Row(n)

    R, x = PolynomialRing(ZZ, "x")

    p = chebyshev_t(n, x)

    [coeff(p, j) for j in 0:n] end

for n in 0:6 A053120Row(n) |> println end # Peter Luschny, Mar 13 2018

CROSSREFS

Cf. A039991, A000012, A001333.

The first nonzero (sub)diagomal sequences are A011782, -A001792, A001793(n+1), -A001794, A006974, -A006975, A006976, -A209404.

Sequence in context: A223707 A046767 A115720 * A336836 A284976 A008743

Adjacent sequences:  A053117 A053118 A053119 * A053121 A053122 A053123

KEYWORD

sign,tabl,nice,easy

AUTHOR

Wolfdieter Lang

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 29 17:21 EST 2020. Contains 338769 sequences. (Running on oeis4.)