OFFSET
0,6
COMMENTS
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.
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 5.
Tom 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.
Wolfdieter Lang, Rows n = 0..20.
Wikipedia, Chebyshev polynomials
FORMULA
T(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).
T(n, m) := 0 if n < m or n+m odd; T(n, m) = (-1)^(n/2) if m=0 (n even); otherwise T(n, m) = ((-1)^((n+m)/2 + m))*(2^(m-1))*n*binomial((n+m)/2-1, m-1)/m.
Recursion for n >= 2: T(n, m) = T*a(n-1, m-1) - T(n-2, m), T(n, m)=0 if n < m, T(n, -1) := 0, T(0, 0) = T(1, 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).
From G. C. Greubel, Aug 10 2022: (Start)
Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n).
T(2*n, n) = i^n * A036909(n/2) * (1+(-1)^n)/2 + [n=0]/3. (End)
T(n, k) = [x^k] T(n, x) for n >= 1, where T(n, x) = Sum_{k=1..n}(-1)^(n - k)*(n/ (2*k))*binomial(k, n - k)*(2*x)^(2*k - n). - Peter Luschny, Sep 20 2022
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
T := (n, x) -> `if`(n = 0, 1, add((-1)^(n - k) * (n/(2*k))*binomial(k, n - k) *(2*x)^(2*k - n), k = 1 ..n)):
seq(seq(coeff(T(n, x), x, k), k = 0..n), n = 0..11); # Peter Luschny, Sep 20 2022
MATHEMATICA
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 *)
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
(SageMath)
def f(n, k): # f = A039991
if (n<2 and k==0): return 1
elif (k<0 or k>n): return 0
else: return 2*f(n-1, k) - f(n-2, k-2)
def A053120(n, k): return f(n, n-k)
flatten([[A053120(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 10 2022
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved