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%I #67 Jun 13 2022 15:21:24
%S 1,1,97,19601,7380481,4517251249,4097989415521,5170128475599457,
%T 8661355881006882817,18605234632923999244961,
%U 49862414878754347585980001,163104845048002042971670685041,639582975902942936737758325440001
%N a(n) = cosh(2 * n * arccosh(n)).
%H Seiichi Manyama, <a href="/A173129/b173129.txt">Table of n, a(n) for n = 0..193</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) = (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
%F a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n^2-1)^(n-k)*n^(2*k). - _Seiichi Manyama_, Dec 27 2018
%F a(n) = T_{2n}(n) where T_{2n} is a Chebyshev polynomial of the first kind. - _Robert Israel_, Dec 27 2018
%F 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
%p seq(orthopoly[T](2*n,n), n=0..50); # _Robert Israel_, Dec 27 2018
%t Table[Round[Cosh[2 n ArcCosh[n]]], {n, 0, 20}] (* _Artur Jasinski_, Feb 10 2010 *)
%t Round[Table[1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x), {x, 0, 10}]] (* _Artur Jasinski_, Feb 14 2010 *)
%t Table[ChebyshevT[2*n, n], {n, 0, 15}] (* _Vaclav Kotesovec_, Nov 07 2021 *)
%o (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
%o (PARI) {a(n) = polchebyshev(2*n, 1, n)} \\ _Seiichi Manyama_, Dec 28 2018
%o (PARI) {a(n) = polchebyshev(n, 1, 2*n^2-1)} \\ _Seiichi Manyama_, Dec 29 2018
%Y Cf. A001079, A037270, A053120 (Chebyshev polynomial), A058331, A115066, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173148.
%Y Cf. A349070, A349071, A349073.
%K nonn
%O 0,3
%A _Artur Jasinski_, Feb 10 2010
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