A322904 - OEIS

%I #26 Sep 08 2022 08:46:23

%S 1,1,181,38081,14526601,8943235489,8138661470941,10287228590683393,

%T 17254778510170993681,37095265466946847758401,

%U 99474891266913130060486021,325534304813775692747248543681,1276941308627620432293188401109401,5914558735952850788377566338591400673

%N a(n) = Sum_{k=0..n} binomial(2*n+1,2*k+1)*(n^2-1)^(n-k)*n^(2*k).

%H Seiichi Manyama, <a href="/A322904/b322904.txt">Table of n, a(n) for n = 0..193</a>

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

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F For n > 0, a(n) = (1/n) * T_{2*n+1}(n) where T_{n}(x) is a Chebyshev polynomial of the first kind.

%F For n > 0, a(n) = (1/n) * cosh((2*n+1)*arccosh(n)).

%F a(n) ~ 4^n * n^(2*n). - _Vaclav Kotesovec_, Jan 03 2019

%t a[0] = 1; a[n_] := 1/n ChebyshevT[2n+1, n];

%t Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Jan 02 2019 *)

%o (PARI) {a(n) = sum(k=0, n, binomial(2*n+1, 2*k+1)*(n^2-1)^(n-k)*n^(2*k))}

%o (PARI) a(n) = if (n==0, 1, polchebyshev(2*n+1, 1, n)/n); \\ _Michel Marcus_, Jan 02 2019

%o (Magma) [&+[Binomial(2*n+1,2*k+1)*(n^2-1)^(n-k)*n^(2*k): k in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Jan 03 2019

%Y Diagonal of A188646.

%Y Cf. A253880, A302329, A302330, A302331, A302332.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 30 2018