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A053117 Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order). 20
1, 0, 2, -1, 0, 4, 0, -4, 0, 8, 1, 0, -12, 0, 16, 0, 6, 0, -32, 0, 32, -1, 0, 24, 0, -80, 0, 64, 0, -8, 0, 80, 0, -192, 0, 128, 1, 0, -40, 0, 240, 0, -448, 0, 256, 0, 10, 0, -160, 0, 672, 0, -1024, 0, 512, -1, 0, 60, 0, -560, 0, 1792, 0, -2304, 0, 1024, 0, -12, 0, 280, 0, -1792, 0, 4608, 0, -5120, 0, 2048, 1, 0, -84, 0, 1120, 0, -5376, 0, 11520 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n,m) = (2^m)*A049310(n,m).

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

Row sums (unsigned triangle) A000129(n+1) (Pell). Row sums (signed triangle) A000027(n+1) (natural numbers).

The o.g.f. for the Legendre polynomials L(n,x) is 1 / sqrt(1- 2x*z + z^2), and squaring it gives the o.g.f. of this entry, so Sum_{k=0..n} L(k,x) L(n-k,x) = U(n,x). This reduces to U(n,x) = L(n/2,x)^2 + 2*Sum_{k=0...n/2-1} L(k,x) L(n-k,x) for n even and U(n,x) = 2*Sum_{k=0..(n-1)/2} L(k,x) L(n-k.x) for odd n. (Cf. also Allouche et al.) For a connection through the Legendre polynomials to elliptic curves and modular forms, see the MathOverflow question below. For the normalized Legendre polynomials, see A100258. (Cf. A097610 with h1 = -2x and h2 = 1, A207538, A099089 and A133156.) - Tom Copeland, Feb 04 2016

The compositional inverse of the shifted o.g.f. x / (1 + 2xz + z^2) for differently signed row polynomials of this entry is the shifted o.g.f. of A121448. The unsigned, non-vanishing antidiagonals (top to bottom) of this triangle are the rows of A038207. - Tom Copeland, Feb 08 2016

REFERENCES

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

J. Allouche and G. Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol. 214, Issue 1-3, 21 March 2000, pp. 21-49.

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

P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). - 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.

MathOverflow, Geometric picture of invariant differential of an elliptic curve, Dec 4 2011.

R. Pemantle and M. C. Wilson, Asymptotics of multivariate sequences, I: smooth points of the singular variety, arXiv:math/0003192 [math.CO], 2000.

A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

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

If n and k are of the same parity then a(n,k)=(-1)^((n-k)/2)*sum(binomial((n+k)/2,i)*binomial((n+k)/2-i,(n-k)/2),i=0..k) and a(n,k)=0 otherwise. - Milan Janjic, Apr 13 2008

EXAMPLE

Triangle begins:

   1;

   0,  2;

  -1,  0,   4;

   0, -4,   0, 8;

   1,  0, -12, 0, 16;

... E.g., fourth row (n=3) {0,-4,0,8} corresponds to polynomial U(3,x) = -4*x + 8*x^3.

MAPLE

seq(seq(coeff(orthopoly[U](n, x), x, j), j=0..n), n=0..16); # Robert Israel, Feb 09 2016

MATHEMATICA

Flatten[ Table[ CoefficientList[ ChebyshevU[n, x], x], {n, 0, 12}]](* Jean-Fran├žois Alcover, Nov 24 2011 *)

PROG

(PARI) T(n, k) = polcoeff(polchebyshev(n, 2), k); \\ Michel Marcus, Feb 10 2016

(Julia)

using Nemo

function A053117Row(n)

    R, x = PolynomialRing(ZZ, "x")

    p = chebyshev_u(n, x)

    [coeff(p, j) for j in 0:n] end

for n in 0:6 A053117Row(n) |> println end # Peter Luschny, Mar 13 2018

CROSSREFS

Cf. A000027, A000129, A049310, A053118.

Cf. A038207, A097610, A099089, A100258, A121448, A133156, A207538.

Sequence in context: A214809 A137336 A115322 * A121448 A019094 A134082

Adjacent sequences:  A053114 A053115 A053116 * A053118 A053119 A053120

KEYWORD

easy,nice,sign,tabl,look

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified May 23 00:47 EDT 2018. Contains 304445 sequences. (Running on oeis4.)