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

Author

Roger L. Bagula, Mar 01 2006

Keywords

• nonn

Examples

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

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.

Crossrefs

Cf. A097690, A323118, A342167.

Programs

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 *)

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

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

Extensions

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