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

1, 2, 17, 244, 4801, 120126, 3650401, 130576328, 5374978561, 250283080090, 13007560326001, 746411226303612, 46873096812360001, 3197490648645613334, 235451028081583642049, 18614381236112230383376, 1572584048032918633353217

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..351

Wikipedia, Chebyshev polynomials.

Formula

a(n) = cos(n*arccos(n+1)).

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

From Peter Bala, Mar 11 2024: (Start)

a(2*n+1) == 1 (mod (2*n + 1)^3); a(2*n) == 1 (mod (n + 1)*(2*n)^3).

a(n) = hypergeom([n, -n], [1/2], -n/2). (End)

a(n) ~ exp(1) * 2^(n-1) * n^n. - Vaclav Kotesovec, Mar 12 2024

Mathematica

Table[ChebyshevT[n, n + 1], {n, 0, 16}] (* Amiram Eldar, Mar 05 2021 *)

Prog

(PARI) a(n) = polchebyshev(n, 1, n+1);
(PARI) a(n) = round(cos(n*acos(n+1)));
(PARI) a(n) = if(n==0, 1, n*sum(k=0, n, (2*n)^k*binomial(n+k, 2*k)/(n+k)));

Crossrefs

Cf. A115066, A323117, A342206, A342207, A370259, A370260, A370261.

Sequence in context: A240999 A319947 A361194 * A099694 A099698 A218681

Adjacent sequences: A342202 A342203 A342204 * A342206 A342207 A342208

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 05 2021

Status

approved