A081265 Triangle of coefficients of the polynomials a(n, x) = 2*a(n-1, x)+ x^2*a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

1, 1, 0, 2, 0, 1, 4, 0, 3, 0, 8, 0, 8, 0, 1, 16, 0, 20, 0, 5, 0, 32, 0, 48, 0, 18, 0, 1, 64, 0, 112, 0, 56, 0, 7, 0, 128, 0, 256, 0, 160, 0, 32, 0, 1, 256, 0, 576, 0, 432, 0, 120, 0, 9, 0, 512, 0, 1280, 0, 1120, 0, 400, 0, 50, 0, 1, 1024, 0, 2816, 0, 2816, 0, 1232, 0, 220

Offset

0,4

Comments

Unsigned Chebyshev numbers of the first kind. Columns include A011782, A001792, A001793, A001794, A006974.

For the Riordan coefficient triangle for Chebyshev's T-polynomials (decreasing odd or even powers of x) see A039991. - Wolfdieter Lang, Aug 06 2014

Links

Table of n, a(n) for n=0..74.

Formula

T(n,k) = [x^k] a(n,x), k = 0, 1, ..., n, with polynomial a(n,x) defined by the recurrence given as name. Its Binet-de Moivre form is a(n, x) = ((1+sqrt(x^2+1))^n + (1-sqrt(x^2+1))^n)/2.

O.g.f. for row polynomials a(n,x): (1-z)/(1 - 2*z - (x*z)^2). Compare with A039991.

Example

Triangle rows are {1}, {1,0}, {2,0,1}, {4,0,3,0}, {8,0,8,0,1},.... [Corrected by Philippe Deléham, Dec 27 2007]
See the unsigned example under A039991. - Wolfdieter Lang, Aug 06 2014

Crossrefs

Cf. A008310, A039991 (signed).

Sequence in context: A261251 A341101 A342909 * A039991 A273821 A363519

Adjacent sequences: A081262 A081263 A081264 * A081266 A081267 A081268

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 15 2003

Extensions

Edited. Name and formula clarified. G.f. of row polynomial, and crossref. A039991 added. - Wolfdieter Lang, Aug 06 2014

Status

approved