A107995 - OEIS

%I #23 Mar 05 2021 10:15:55

%S 1,6,63,980,20305,526890,16451071,600940872,25154396001,1187422368110,

%T 62418042417599,3616337930622300,228977061309711793,

%U 15731733543660288210,1165677769357309014015,92665403695822344828176

%N Chebyshev polynomial of the second kind U[n,x] evaluated at x=n+2.

%D Rosenblum and Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18.

%D G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, 1939, p. 29.

%D G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1966, p. 35.

%H Seiichi Manyama, <a href="/A107995/b107995.txt">Table of n, a(n) for n = 0..351</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

%F a(n) = Sum_{k=0..n} (2*n+2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (2*n+2)^k * binomial(n+1+k,2*k+1). - _Seiichi Manyama_, Mar 05 2021

%F a(n) ~ exp(2) * 2^n * n^n. - _Vaclav Kotesovec_, Mar 05 2021

%e a(3)=980 because U[3,x]=8x^3-4x and U[3,5]=8*5^3-4*5=980.

%p with(orthopoly): seq(U(n,n+2),n=0..17);

%t Table[ChebyshevU[n, n + 2], {n, 0, 15}] (* _Amiram Eldar_, Mar 05 2021 *)

%o (PARI) a(n) = polchebyshev(n, 2, n+2); \\ _Seiichi Manyama_, Mar 05 2021

%o (PARI) a(n) = sum(k=0, n, (2*n+2)^k*binomial(n+1+k, 2*k+1)); \\ _Seiichi Manyama_, Mar 05 2021

%Y Cf. A097690, A323118, A342167.

%K nonn

%O 0,2

%A _Roger L. Bagula_, Mar 01 2006

%E Edited by _N. J. A. Sloane_, Apr 05 2006