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A342205 a(n) = T(n,n+1) where T(n,x) is a Chebyshev polynomial of the first kind.

%I A342205 #30 Mar 12 2024 10:54:00
%S A342205 1,2,17,244,4801,120126,3650401,130576328,5374978561,250283080090,
%T A342205 13007560326001,746411226303612,46873096812360001,3197490648645613334,
%U A342205 235451028081583642049,18614381236112230383376,1572584048032918633353217
%N A342205 a(n) = T(n,n+1) where T(n,x) is a Chebyshev polynomial of the first kind.
%H A342205 Seiichi Manyama, <a href="/A342205/b342205.txt">Table of n, a(n) for n = 0..351</a>
%H A342205 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%F A342205 a(n) = cos(n*arccos(n+1)).
%F A342205 a(n) = n * Sum_{k = 0..n} (2*n)^k * binomial(n+k,2*k)/(n+k) for n > 0.
%F A342205 From _Peter Bala_, Mar 11 2024: (Start)
%F A342205 a(2*n+1) == 1 (mod (2*n + 1)^3); a(2*n) == 1 (mod (n + 1)*(2*n)^3).
%F A342205 a(n) = hypergeom([n, -n], [1/2], -n/2). (End)
%F A342205 a(n) ~ exp(1) * 2^(n-1) * n^n. - _Vaclav Kotesovec_, Mar 12 2024
%t A342205 Table[ChebyshevT[n, n + 1], {n, 0, 16}] (* _Amiram Eldar_, Mar 05 2021 *)
%o A342205 (PARI) a(n) = polchebyshev(n, 1, n+1);
%o A342205 (PARI) a(n) = round(cos(n*acos(n+1)));
%o A342205 (PARI) a(n) = if(n==0, 1, n*sum(k=0, n, (2*n)^k*binomial(n+k, 2*k)/(n+k)));
%Y A342205 Cf. A115066, A323117, A342206, A342207, A370259, A370260, A370261.
%K A342205 nonn,easy
%O A342205 0,2
%A A342205 _Seiichi Manyama_, Mar 05 2021