|
| |
|
|
A053120
|
|
Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order).
|
|
163
|
|
|
|
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).
|
|
|
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.
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; http://repository.wit.ie/1693/1/AoifeThesis.pdf
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.
W. 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
|
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;
... E.g. fourth row (n=3) corresponds to polynomial T(3,x)= -3*x+4*x^3.
|
|
|
MATHEMATICA
|
t[n_, k_] := Coefficient[ ChebyshevT[n, x], x, k]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* From Jean-François 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
|
easy,nice,sign,tabl
|
|
|
AUTHOR
|
Wolfdieter Lang
|
|
|
STATUS
|
approved
|
| |
|
|
