A323012 a(n) = (1/sqrt(n^2+1)) * T_{2*n+1}(sqrt(n^2+1)) where T_{n}(x) is a Chebyshev polynomial of the first kind.

1, 5, 305, 53353, 18674305, 10928351501, 9616792908241, 11868363584907985, 19553538801258341377, 41456387654578883552149, 109939727677547706703222001, 356521758767660233608385698361, 1387930545993760882531890016305025

Offset: 0

Author

Seiichi Manyama, Jan 02 2019

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 A188647.

Programs

Magma

[&+[Binomial(2*n+1,2*k)*(n^2+1)^(n-k)*n^(2*k): k in [0..n]]: n in [0..15]]; // Vincenzo Librandi, Jan 03 2019

Mathematica

Join[{1}, Table[Sum[Binomial[2 n + 1, 2 k] (n^2 + 1)^(n - k) n^(2 k), {k, 0, n}], {n, 20}]] (* Vincenzo Librandi, Jan 03 2019 *)

PARI

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

Formula

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

a(n) ~ 2^(2*n) * n^(2*n). - Vaclav Kotesovec, Jan 02 2019