A342206 - OEIS

%I #23 Mar 12 2024 09:38:17

%S 1,3,31,485,10081,262087,8193151,299537289,12545596801,592479412811,

%T 31154649926687,1805486216133613,114342125644787041,

%U 7857107443850071695,582268591681887560191,46292552162781456490001,3930448770533424343942657

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

%H Seiichi Manyama, <a href="/A342206/b342206.txt">Table of n, a(n) for n = 0..351</a>

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

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

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

%F a(n) ~ exp(2) * 2^(n-1) * n^n. - _Vaclav Kotesovec_, Mar 12 2024

%t Table[ChebyshevT[n, n + 2], {n, 0, 16}] (* _Amiram Eldar_, Mar 05 2021 *)

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

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

%o (PARI) a(n) = if(n==0, 1, n*sum(k=0, n, (2*n+2)^k*binomial(n+k, 2*k)/(n+k)));

%Y Cf. A107995, A115066, A323117, A342205.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 05 2021