A053124 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).

1, -2, 4, 3, -16, 16, -4, 40, -96, 64, 5, -80, 336, -512, 256, -6, 140, -896, 2304, -2560, 1024, 7, -224, 2016, -7680, 14080, -12288, 4096, -8, 336, -4032, 21120, -56320, 79872, -57344, 16384, 9, -480, 7392, -50688, 183040, -372736, 430080, -262144, 65536, -10, 660, -12672, 109824, -512512, 1397760, -2293760, 2228224

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nice
• sign
• tabl

Comments

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

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.

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

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.

Examples

			{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.
		

References

• C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
• G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, problem 39, page 7.
• 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..50 of triangle, flattened
• C. Lanczos, Applied Analysis (Annotated scans of selected pages). See page 518.
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A053122, A053125.

Programs

Mathematica

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

Formula

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

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;

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

In other words, Riordan array (1/(1+x)^2, 4x/(1+x)^2). - Ralf Stephan, Jan 21 2014