0,2

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.

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

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

T. D. Noe, Rows n=0..50 of triangle, flattened

Index entries for sequences related to Chebyshev polynomials.

a(n, m) := 0 if n<m else (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.

{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.

Cf. A053122, A053125.

Sequence in context: A115399 A109429 A114894 this_sequence A071970 A163089 A111172

Adjacent sequences: A053121 A053122 A053123 this_sequence A053125 A053126 A053127

easy,nice,sign,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)