A349073 a(n) = U(2*n, n), where U(n, x) is the Chebyshev polynomial of the second kind.

1, 3, 209, 40391, 15003009, 9127651499, 8254109243953, 10393834843080975, 17391182043967249409, 37326390852372133364819, 99976027392046047055178001, 326887883645157139828711692503, 1281398359905415379814555044995201, 5932135472283024519893762690145006075

Offset: 0

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..193
• Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind.
• Wikipedia, Chebyshev polynomials.

Crossrefs

Cf. A173129, A323118, A349071, A349074, A349075.

Programs

Mathematica

Table[ChebyshevU[2*n, n], {n, 0, 15}]

PARI

a(n) = polchebyshev(2*n, 2, n); \\ Michel Marcus, Nov 07 2021

Python

from sympy import chebyshevu
def A349073(n): return chebyshevu(n<<1,n) # Chai Wah Wu, Nov 08 2023

Formula

For n>1, a(n) = ((n + sqrt(n^2-1))^(2*n+1) - (n - sqrt(n^2-1))^(2*n+1)) / (2*sqrt(n^2-1)).

a(n) ~ 2^(2*n) * n^(2*n).