%I #28 Dec 25 2019 08:32:14

%S 1,-2,4,3,-16,16,-4,40,-96,64,5,-80,336,-512,256,-6,140,-896,2304,

%T -2560,1024,7,-224,2016,-7680,14080,-12288,4096,-8,336,-4032,21120,

%U -56320,79872,-57344,16384,9,-480,7392,-50688,183040,-372736,430080,-262144,65536,-10,660,-12672,109824,-512512,1397760,-2293760,2228224

%N Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).

%C a(n,m) = (4^m)*A053122(n,m).

%C G.f. for row polynomials U^{*}(n,x) = U(n,2*x-1) (signed triangle): 1/(1+2*z*(1-2*x) + z^2). Unsigned triangle |a(n,m)| has g.f. 1/(1-2*z*(1+2*x)+z^2) for the row polynomials.

%C Row sums (signed triangle) A000027(n+1) (natural numbers). Row sums (unsigned triangle) A001109(n+1).

%C In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to a Riordan group.

%D C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.

%D G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, problem 39, page 7.

%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

%H T. D. Noe, <a href="/A053124/b053124.txt">Rows n = 0..50 of triangle, flattened</a>

%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages). See page 518.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F a(n, m) := 0 if n < m, otherwise (4^m)*((-1)^(n-m))*binomial(n+m+1, 2*m+1);

%F a(n, m) = -2*a(n-1, m) + 4*a(n-1, m-1) - a(n-2, m), a(n, m) := 0 if n=-1 or m=-1 or n < m, a(0, 0)=1;

%F g.f. for m-th column (signed triangle): ((4*x/(1+x)^2)^m)/(1+x)^2.

%F In other words, Riordan array (1/(1+x)^2, 4x/(1+x)^2). - _Ralf Stephan_, Jan 21 2014

%e {1}; {-2,4}; {3,-16,16}; {-4,40,-96,64}; {5,-80,336,-512,256};... E.g., fourth row (n=3) {-4,40,-96,64} corresponds to polynomial U(3,2*x-1)= -4+40*x-96*x^2+64*x^3.

%t Table[ CoefficientList[ ChebyshevU[n, 2x - 1], x], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Dec 05 2012 *)

%Y Cf. A053122, A053125.

%K easy,nice,sign,tabl

%O 0,2

%A _Wolfdieter Lang_