A349073 - OEIS

%I #19 Nov 08 2023 11:32:30

%S 1,3,209,40391,15003009,9127651499,8254109243953,10393834843080975,

%T 17391182043967249409,37326390852372133364819,

%U 99976027392046047055178001,326887883645157139828711692503,1281398359905415379814555044995201,5932135472283024519893762690145006075

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

%H Seiichi Manyama, <a href="/A349073/b349073.txt">Table of n, a(n) for n = 0..193</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html">Chebyshev Polynomial of the Second Kind</a>.

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

%F 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)).

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

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

%o (PARI) a(n) = polchebyshev(2*n, 2, n); \\ _Michel Marcus_, Nov 07 2021

%o (Python)

%o from sympy import chebyshevu

%o def A349073(n): return chebyshevu(n<<1,n) # _Chai Wah Wu_, Nov 08 2023

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

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 07 2021