OFFSET
0,3
COMMENTS
The Chebyshev polynomial T_n is defined by cos(nx) = T_n(cos(x)). So T_2n(cos(x)) = cos(2nx) = cos^2(nx) - 1 = (T_n(x))^2 - 1 consists of only even powers of x. As a result, a(n) = T_2n(sqrt(n)) is an integer. - Michael B. Porter, Jan 01 2019
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..321
Wikipedia, Chebyshev polynomials.
FORMULA
a(n) ~ exp(-1/2) * 2^(2*n-1) * n^n. - Vaclav Kotesovec, Apr 05 2016
a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n-1)^(n-k)*n^k. - Seiichi Manyama, Dec 27 2018
a(n) = cosh(2*n*arccosh(sqrt(n))). - Seiichi Manyama, Dec 27 2018
a(n) = T_{2*n}(sqrt(n)) = T_{n}(2*n-1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018
a(n) = A322790(n-1, n) for n > 0. - Seiichi Manyama, Dec 29 2018
MATHEMATICA
Table[Round[Cos[2 n ArcCos[Sqrt[n]]]], {n, 0, 30}] (* Artur Jasinski, Feb 11 2010 *)
PROG
(PARI) {a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n-1)^(n-k)*n^k)} \\ Seiichi Manyama, Dec 27 2018
(PARI) {a(n) = round(cosh(2*n*acosh(sqrt(n))))} \\ Seiichi Manyama, Dec 27 2018
(PARI) {a(n) = polchebyshev(n, 1, 2*n-1)} \\ Seiichi Manyama, Dec 29 2018
(GAP) a:=List([0..20], n->Sum([0..n], k->Binomial(2*n, 2*k)*(n-1)^(n-k)*n^k));; Print(a); # Muniru A Asiru, Jan 03 2019
(Magma) [&+[Binomial(2*n, 2*k)*(n-1)^(n-k)*n^k: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Jan 03 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Feb 11 2010
EXTENSIONS
Minor edits by Vaclav Kotesovec, Apr 05 2016
STATUS
approved