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URL: https://oeis.org/A342205

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A342205
a(n) = T(n,n+1) where T(n,x) is a Chebyshev polynomial of the first kind.
8
1, 2, 17, 244, 4801, 120126, 3650401, 130576328, 5374978561, 250283080090, 13007560326001, 746411226303612, 46873096812360001, 3197490648645613334, 235451028081583642049, 18614381236112230383376, 1572584048032918633353217
OFFSET
0,2
LINKS
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)));
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Mar 05 2021
STATUS
approved