cp's OEIS Frontend

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

%I A115066 #49 Feb 05 2022 11:22:25
%S A115066 1,1,7,99,1921,47525,1431431,50843527,2081028097,96450076809,
%T A115066 4993116004999,285573847759211,17882714781360001,1216895030905226413,
%U A115066 89415396036432386311,7055673735003659189775,595077930963909484707841,53421565080956452077519377
%N A115066 Chebyshev polynomial of the first kind T(n,x), evaluated at x=n.
%D A115066 G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1966, p. 35.
%D A115066 M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 18.
%D A115066 G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, 1939, p. 29.
%H A115066 Seiichi Manyama, <a href="/A115066/b115066.txt">Table of n, a(n) for n = 0..351</a>
%F A115066 a(n) = cos(n*arccos(n)).
%F A115066 a(n) ~ 2^(n-1) * n^n. - _Vaclav Kotesovec_, Jan 19 2019
%F A115066 a(n) = (A323118(n) - A107995(n-2))/2 for n > 1. - _Seiichi Manyama_, Mar 05 2021
%F A115066 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
%F A115066 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
%e A115066 a(3) = 99 because T[3, x] = 4x^3 - 3x and T[3, 3] = 4*3^3 - 3*3 = 99.
%p A115066 with(orthopoly): seq(T(n,n),n=0..17);
%t A115066 Table[ChebyshevT[n, n], {n, 0, 17}] (* _Arkadiusz Wesolowski_, Nov 17 2012 *)
%o A115066 (PARI) A115066(n)=cos(n*acos(n))  \\ _M. F. Hasler_, Apr 06 2012
%o A115066 (PARI) a(n) = polchebyshev(n, 1, n); \\ _Seiichi Manyama_, Dec 28 2018
%o A115066 (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
%Y A115066 Main diagonal of A322836.
%Y A115066 Cf. A053120, A107995, A173129, A323118, A349070, A349071, A349072.
%K A115066 nonn,easy
%O A115066 0,3
%A A115066 _Roger L. Bagula_, Mar 01 2006
%E A115066 Edited by _N. J. A. Sloane_, Apr 05 2006