A053117 - cp's OEIS frontend

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, 0, -11264, 0, 4096

Offset: 0

Wolfdieter Lang

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

			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.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 22, page 196.

T. D. Noe, Rows n=0..100 of triangle, flattened
J.-P. 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.
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.
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.
Valentin Ovsienko, Towards quantized complex numbers: q-deformed Gaussian integers and the Picard group, arXiv:2103.10800 [math.QA], 2021.
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.

Cf. A000027, A000129, A049310, A053118.

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

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

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

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

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

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

a(n, m) := 0 if n

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

cp's OEIS Frontend

A053117 Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).

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