OFFSET
0,2
LINKS
Robert Israel, Table of n, a(n) for n = 0..192
Wikipedia, Chebyshev polynomials.
FORMULA
a(n) = (1/2)*((n + sqrt(n^2 + 1))^(2*n) + (n - sqrt(n^2 + 1))^(2*n)). - Artur Jasinski, Feb 14 2010, corrected by Vaclav Kotesovec, Apr 05 2016
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n^2+1)^(n-k)*n^(2*k). - Seiichi Manyama, Dec 27 2018
a(n) = T_{n}(2*n^2+1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018
MAPLE
seq(expand( (1/2)*((n + sqrt(n^2 + 1))^(2*n) + (n - sqrt(n^2 + 1))^(2*n))), n=0..30); # Robert Israel, Apr 05 2016
MATHEMATICA
Round[Table[Cosh[2 n ArcSinh[n]], {n, 0, 20}]] (* Artur Jasinski *)
Round[Table[1/2 (x - Sqrt[1 + x^2])^(2 x) + 1/2 (x + Sqrt[1 + x^2])^(2 x), {x, 0, 20}]] (* Artur Jasinski, Feb 14 2010 *)
PROG
(PARI) {a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2+1)^(n-k)*n^(2*k))} \\ Seiichi Manyama, Dec 27 2018
(PARI) {a(n) = polchebyshev(n, 1, 2*n^2+1)} \\ Seiichi Manyama, Dec 29 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Feb 10 2010
STATUS
approved