login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A053119 Triangle of coefficients of Chebyshev's S(n,x) polynomials (exponents in decreasing order). 9
1, 1, 0, 1, 0, -1, 1, 0, -2, 0, 1, 0, -3, 0, 1, 1, 0, -4, 0, 3, 0, 1, 0, -5, 0, 6, 0, -1, 1, 0, -6, 0, 10, 0, -4, 0, 1, 0, -7, 0, 15, 0, -10, 0, 1, 1, 0, -8, 0, 21, 0, -20, 0, 5, 0, 1, 0, -9, 0, 28, 0, -35, 0, 15, 0, -1, 1, 0, -10, 0, 36, 0, -56, 0, 35, 0, -6, 0, 1, 0, -11, 0, 45, 0, -84, 0, 70, 0, -21, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

a(n,m) = A049310(n,n-m).

G.f. for row polynomials S(n,x) (signed triangle): 1/(1-x*z+z^2). Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,x) as row polynomials with G.f. 1/(1-x*z-z^2).

REFERENCES

D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 232, Sect. 3.3.38.

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

LINKS

T. D. Noe, Rows n=0..100 of triangle, flattened

T. Copeland, Addendum to Elliptic Lie Triad

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n, m) := 0 if n<m or m odd, else ((-1)^(3*m/2))*binomial(n-m/2, n-m); a(n, m) = a(n-1, m) - a(n-2, m-2), a(n, -2) := 0 =: a(n, -1), a(0, 0)=1, a(n, m)= 0 if n<m or m odd; G.f. for m-th column (signed triangle): (-1)^(3*m/2)*x^m/(1-x)^(m/2+1) if m >= 0 is even else 0.

Recurrence for the (unsigned) Fibonacci polynomials: F[1]=1, F[2]=x; for n>2, F[n] = x*F[n-1]+F[n-2].

EXAMPLE

The triangle begins:

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

0:  1

1:  1  0

2:  1  0  -1

3:  1  0  -2  0

4:  1  0  -3  0   1

5:  1  0  -4  0   3  0

6:  1  0  -5  0   6  0  -1

7:  1  0  -6  0  10  0  -4  0

8:  1  0  -7  0  15  0 -10  0   1

9:  1  0  -8  0  21  0 -20  0   5  0

10: 1  0  -9  0  28  0 -35  0  15  0  -1

... Reformatted. - Wolfdieter Lang, Dec 17 2013

E.g., fourth row (n=3) corresponds to polynomial S(3,x)= x^3-2*x.

Triangle of absolute values of coefficients (coefficients of Fibonacci polynomials) with exponents in increasing order begins:

[1]

[0, 1]

[1, 0, 1]

[0, 2, 0, 1]

[1, 0, 3, 0, 1]

[0, 3, 0, 4, 0, 1]

[1, 0, 6, 0, 5, 0, 1]

[0, 4, 0, 10, 0, 6, 0, 1]

[1, 0, 10, 0, 15, 0, 7, 0, 1]

[0, 5, 0, 20, 0, 21, 0, 8, 0, 1]

See A162515 for the Fibonacci polynomials with reversed row entries, starting there with row 1. - Wolfdieter Lang, Dec 16 2013

MATHEMATICA

ChebyshevS[n_, x_] := ChebyshevU[n, x/2]; Flatten[ Table[ Reverse[ CoefficientList[ ChebyshevS[n, x], x]], {n, 0, 12}]] (* Jean-Fran├žois Alcover, Nov 25 2011 *)

PROG

(PARI) tabl(nn) = for (n=0, nn, print(Vec(polchebyshev(n, 2, x/2)))); \\ Michel Marcus, Jan 14 2016

CROSSREFS

Row sums give A000045. Reflection of A049310.

Cf. A162515. - Wolfdieter Lang, Dec 16 2013

Sequence in context: A225345 A083280 A060689 * A162515 A175267 A108045

Adjacent sequences:  A053116 A053117 A053118 * A053120 A053121 A053122

KEYWORD

easy,nice,sign,tabl

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 25 22:44 EST 2018. Contains 299662 sequences. (Running on oeis4.)