OFFSET
0,3
REFERENCES
G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1966, p. 35.
M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 18.
G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, 1939, p. 29.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..351
FORMULA
a(n) = cos(n*arccos(n)).
a(n) ~ 2^(n-1) * n^n. - Vaclav Kotesovec, Jan 19 2019
a(n) = n * Sum_{k=0..n} (2*n-2)^k * binomial(n+k,2*k)/(n+k) for n > 0. - Seiichi Manyama, Mar 05 2021
It appears that a(2*n+1) == 0 (mod (2*n+1)^2) and 2*a(4*n+2) == -2 (mod (4*n+2)^4), while for k > 1, 2*a(2^k*(2*n+1)) == 2 (mod (2^k*(2*n+1))^4). - Peter Bala, Feb 01 2022
EXAMPLE
a(3) = 99 because T[3, x] = 4x^3 - 3x and T[3, 3] = 4*3^3 - 3*3 = 99.
MAPLE
with(orthopoly): seq(T(n, n), n=0..17);
MATHEMATICA
Table[ChebyshevT[n, n], {n, 0, 17}] (* Arkadiusz Wesolowski, Nov 17 2012 *)
PROG
(PARI) A115066(n)=cos(n*acos(n)) \\ M. F. Hasler, Apr 06 2012
(PARI) a(n) = polchebyshev(n, 1, n); \\ Seiichi Manyama, Dec 28 2018
(PARI) a(n) = if(n==0, 1, n*sum(k=0, n, (2*n-2)^k*binomial(n+k, 2*k)/(n+k))); \\ Seiichi Manyama, Mar 05 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Mar 01 2006
EXTENSIONS
Edited by N. J. A. Sloane, Apr 05 2006
STATUS
approved