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

%I A053117 #91 Dec 30 2024 11:27:03
%S A053117 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,
%T A053117 0,-8,0,80,0,-192,0,128,1,0,-40,0,240,0,-448,0,256,0,10,0,-160,0,672,
%U A053117 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
%N A053117 Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).
%C A053117 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).
%C A053117 Row sums (unsigned triangle) A000129(n+1) (Pell). Row sums (signed triangle) A000027(n+1) (natural numbers).
%C A053117 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
%C A053117 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
%D A053117 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
%D A053117 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 22, page 196.
%H A053117 T. D. Noe, <a href="/A053117/b053117.txt">Rows n=0..100 of triangle, flattened</a>
%H A053117 J.-P. Allouche and G. Skordev, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00195-8">Schur congruences, Carlitz sequences of polynomials and automaticity</a>, Discrete Mathematics, Vol. 214, Issue 1-3, 21 March 2000, pp. 21-49.
%H A053117 Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry5/barry96s.html">Meixner-Type Results for Riordan Arrays and Associated Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.9.4, section 5.
%H A053117 P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). - From _Tom Copeland_, Oct 11 2014
%H A053117 Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H A053117 MathOverflow, <a href="http://mathoverflow.net/questions/82597/geometric-picture-of-invariant-differential-of-an-elliptic-curve">Geometric picture of invariant differential of an elliptic curve</a>, Dec 4 2011.
%H A053117 Valentin Ovsienko, <a href="https://arxiv.org/abs/2103.10800">Towards quantized complex numbers: q-deformed Gaussian integers and the Picard group</a>, arXiv:2103.10800 [math.QA], 2021.
%H A053117 R. Pemantle and M. C. Wilson, <a href="http://arXiv.org/abs/math.CO/0003192">Asymptotics of multivariate sequences, I: smooth points of the singular variety</a>, arXiv:math/0003192 [math.CO], 2000.
%H A053117 A. Sapounakis, I. Tasoulas, and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.005">Counting strings in Dyck paths</a>, Discrete Math., 307 (2007), 2909-2924.
%H A053117 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A053117 a(n, m) = (2^m)*A049310(n,m).
%F A053117 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.
%F A053117 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
%e A053117 Triangle begins:
%e A053117    1;
%e A053117    0,  2;
%e A053117   -1,  0,   4;
%e A053117    0, -4,   0, 8;
%e A053117    1,  0, -12, 0, 16;
%e A053117   ...
%e A053117 E.g., fourth row (n=3) {0,-4,0,8} corresponds to polynomial U(3,x) = -4*x + 8*x^3.
%p A053117 seq(seq(coeff(orthopoly[U](n,x),x,j),j=0..n),n=0..16); # _Robert Israel_, Feb 09 2016
%t A053117 Flatten[ Table[ CoefficientList[ ChebyshevU[n, x], x], {n, 0, 12}]](* _Jean-François Alcover_, Nov 24 2011 *)
%o A053117 (PARI) T(n, k) = polcoeff(polchebyshev(n,2), k); \\ _Michel Marcus_, Feb 10 2016
%o A053117 (Julia)
%o A053117 using Nemo
%o A053117 function A053117Row(n)
%o A053117     R, x = PolynomialRing(ZZ, "x")
%o A053117     p = chebyshev_u(n, x)
%o A053117     [coeff(p, j) for j in 0:n] end
%o A053117 for n in 0:6 A053117Row(n) |> println end # _Peter Luschny_, Mar 13 2018
%Y A053117 Cf. A000027, A000129, A049310, A053118.
%Y A053117 Cf. A038207, A097610, A099089, A100258, A121448, A133156, A207538.
%K A053117 easy,nice,sign,tabl,look
%O A053117 0,3
%A A053117 _Wolfdieter Lang_