0,3

a(n,m)= A053122(n,n-m).

G.f. for row polynomials and row sums same as in A053122.

Unsigned column sequences are A000012, A005843, A014105, A002492 for m=0..3, resp. and A053126-A053131 for m=4..9.

This is also the coeficient triangle for Chebyshev's U(2*n+1,x) polynomials expanded in decreasing odd powers of (2*x): U(2*n+1,x)=sum(a(n,m)*(2*x)^(2*(n-m)+1), m=0..n). See the W. Lang link given in A053125.

Unsigned version is mirror image of A078812 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795

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

Stephen Barnett, "Matrices: Methods and Applications", Oxford University Press, 1990, p. 132, 343.

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Index entries for sequences related to Chebyshev polynomials.

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

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

G.f. for m-th column (signed triangle): ((-1)^m)*x^m*Po(m+1, x)/(1-x)^(m+1), with Po(k, x) := sum('binomial(k, 2*j+1)*x^j', 'j'=0..floor(k/2)).

The n-th degree polynomial is the characteristic equation for an n X n tridiagonal matrix with (diagonal = all 2's, sub and superdiagonals all -1's and the rest 0's), exemplified by the 4X4 matrix M = [2 -1 0 0 / -1 2 -1 0 / 0 -1 2 -1 / 0 0 -1 2]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 05 2005

{1}; {1,-2}; {1,-4,3}; {1,-6,10,-4}; {1,-8,21,-20,5};... E.g. fourth row (n=3) {1,-6,10,-4} corresponds to polynomial S(3,x-2)= x^3-6*x^2+10*x-4.

Cf. A053124, A053122.

Sequence in context: A093190 A132191 A094437 this_sequence A107661 A126570 A048790

Adjacent sequences: A053120 A053121 A053122 this_sequence A053124 A053125 A053126

easy,sign,tabl

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