A107995 Chebyshev polynomial of the second kind U[n,x] evaluated at x=n+2.

1, 6, 63, 980, 20305, 526890, 16451071, 600940872, 25154396001, 1187422368110, 62418042417599, 3616337930622300, 228977061309711793, 15731733543660288210, 1165677769357309014015, 92665403695822344828176

Offset

0,2

References

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

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

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..351

Wikipedia, Chebyshev polynomials.

Formula

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

a(n) ~ exp(2) * 2^n * n^n. - Vaclav Kotesovec, Mar 05 2021

Example

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

Maple

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

Mathematica

Table[ChebyshevU[n, n + 2], {n, 0, 15}] (* Amiram Eldar, Mar 05 2021 *)

Prog

(PARI) a(n) = polchebyshev(n, 2, n+2); \\ Seiichi Manyama, Mar 05 2021
(PARI) a(n) = sum(k=0, n, (2*n+2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 05 2021

Crossrefs

Cf. A097690, A323118, A342167.

Sequence in context: A227278 A295267 A098342 * A166893 A213644 A280476

Adjacent sequences: A107992 A107993 A107994 * A107996 A107997 A107998

Keyword

nonn

Author

Roger L. Bagula, Mar 01 2006

Extensions

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

Status

approved