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A173128 a(n) = cosh(2*n*arcsinh(n)). 13

%I

%S 1,3,161,27379,9478657,5517751251,4841332221601,5964153172084899,

%T 9814664424981012481,20791777842234580902499,

%U 55106605639755476546020001,178627672869645203363556318483,695165908550906808156689590141441

%N a(n) = cosh(2*n*arcsinh(n)).

%H Robert Israel, <a href="/A173128/b173128.txt">Table of n, a(n) for n = 0..192</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_{n}(2*n^2+1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - _Seiichi Manyama_, Dec 29 2018

%p 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

%t Round[Table[Cosh[2 n ArcSinh[n]], {n, 0, 20}]] (* _Artur Jasinski_ *)

%t 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 *)

%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(n, 1, 2*n^2+1)} \\ _Seiichi Manyama_, Dec 29 2018

%Y Cf. A058331, A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173129, A173174.

%K nonn

%O 0,2

%A _Artur Jasinski_, Feb 10 2010

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Last modified April 10 22:05 EDT 2021. Contains 342856 sequences. (Running on oeis4.)