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A107995
Chebyshev polynomial of the second kind U[n,x] evaluated at x=n+2.
4
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
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
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Mar 01 2006
EXTENSIONS
Edited by N. J. A. Sloane, Apr 05 2006
STATUS
approved