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a(n) = 2 * T(n,(n+2)/2) where T(n,x) is a Chebyshev polynomial of the first kind.
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%I #18 Apr 11 2021 09:23:23

%S 2,3,14,110,1154,15127,238142,4379769,92198402,2186871698,57721023502,

%T 1678243366813,53301709843202,1836220544383695,68200709735854334,

%U 2716906424134261502,115561578124838522882,5227260815326346060059,250566480717349417632398

%N a(n) = 2 * T(n,(n+2)/2) where T(n,x) is a Chebyshev polynomial of the first kind.

%H Seiichi Manyama, <a href="/A343261/b343261.txt">Table of n, a(n) for n = 0..386</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

%F a(n) = 2 * cos(n*arccos((n+2)/2)).

%F a(n) = 2 * n * Sum_{k=0..n} n^k * binomial(n+k,2*k)/(n+k) for n > 0.

%F a(n) ~ exp(2) * n^n. - _Vaclav Kotesovec_, Apr 09 2021

%t Table[2*ChebyshevT[n, (n+2)/2], {n, 0, 18}] (* _Amiram Eldar_, Apr 09 2021 *)

%o (PARI) a(n) = 2*polchebyshev(n, 1, (n+2)/2);

%o (PARI) a(n) = round(2*cos(n*acos((n+2)/2)));

%o (PARI) a(n) = if(n==0, 2, 2*n*sum(k=0, n, n^k*binomial(n+k, 2*k)/(n+k)));

%Y Main diagonal of A299741.

%Y Cf. A115066, A342167, A342206, A343259, A343260.

%K nonn

%O 0,1

%A _Seiichi Manyama_, Apr 09 2021