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%I #62 Sep 08 2022 08:45:50
%S 1,1,17,485,18817,930249,55989361,3974443213,325142092801,
%T 30122754096401,3117419602578001,356452534779818421,
%U 44627167107085622401,6071840759403431812825,892064955046043465408177,140751338790698080509966749,23737154316161495960243527681
%N a(n) = cos(2*n*arccos(sqrt(n))).
%C 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
%H Seiichi Manyama, <a href="/A173148/b173148.txt">Table of n, a(n) for n = 0..321</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) ~ exp(-1/2) * 2^(2*n-1) * n^n. - _Vaclav Kotesovec_, Apr 05 2016
%F a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n-1)^(n-k)*n^k. - _Seiichi Manyama_, Dec 27 2018
%F a(n) = cosh(2*n*arccosh(sqrt(n))). - _Seiichi Manyama_, Dec 27 2018
%F 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
%F a(n) = A322790(n-1, n) for n > 0. - _Seiichi Manyama_, Dec 29 2018
%t Table[Round[Cos[2 n ArcCos[Sqrt[n]]]], {n, 0, 30}] (* _Artur Jasinski_, Feb 11 2010 *)
%o (PARI) {a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n-1)^(n-k)*n^k)} \\ _Seiichi Manyama_, Dec 27 2018
%o (PARI) {a(n) = round(cosh(2*n*acosh(sqrt(n))))} \\ _Seiichi Manyama_, Dec 27 2018
%o (PARI) {a(n) = polchebyshev(n, 1, 2*n-1)} \\ _Seiichi Manyama_, Dec 29 2018
%o (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
%o (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
%Y Cf. A053120 (Chebyshev polynomial), A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A322790.
%K nonn
%O 0,3
%A _Artur Jasinski_, Feb 11 2010
%E Minor edits by _Vaclav Kotesovec_, Apr 05 2016
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