A115066 Chebyshev polynomial of the first kind T(n,x), evaluated at x=n.

1, 1, 7, 99, 1921, 47525, 1431431, 50843527, 2081028097, 96450076809, 4993116004999, 285573847759211, 17882714781360001, 1216895030905226413, 89415396036432386311, 7055673735003659189775, 595077930963909484707841, 53421565080956452077519377

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) = (A323118(n) - A107995(n-2))/2 for n > 1. - Seiichi Manyama, Mar 05 2021

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

Main diagonal of A322836.

Cf. A053120, A107995, A173129, A323118, A349070, A349071, A349072.

Sequence in context: A238472 A041087 A041084 * A340887 A272957 A123616

Adjacent sequences: A115063 A115064 A115065 * A115067 A115068 A115069

Keyword

nonn,easy

Author

Roger L. Bagula, Mar 01 2006

Extensions

Edited by N. J. A. Sloane, Apr 05 2006

Status

approved