A322904 a(n) = Sum_{k=0..n} binomial(2*n+1,2*k+1)*(n^2-1)^(n-k)*n^(2*k).

1, 1, 181, 38081, 14526601, 8943235489, 8138661470941, 10287228590683393, 17254778510170993681, 37095265466946847758401, 99474891266913130060486021, 325534304813775692747248543681, 1276941308627620432293188401109401, 5914558735952850788377566338591400673

Offset: 0

Author

Seiichi Manyama, Dec 30 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..193
• Wikipedia, Chebyshev polynomials.
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Diagonal of A188646.

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

Programs

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

Mathematica

a[0] = 1; a[n_] := 1/n ChebyshevT[2n+1, n];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Jan 02 2019 *)

PARI

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

PARI

a(n) = if (n==0, 1, polchebyshev(2*n+1, 1, n)/n); \\ Michel Marcus, Jan 02 2019

Formula

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

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

a(n) ~ 4^n * n^(2*n). - Vaclav Kotesovec, Jan 03 2019