A349074 - OEIS

%I #22 Feb 16 2025 08:34:02

%S 1,4,2911,7997214,57641556673,867583274883920,23630375698358890319,

%T 1056918444955456528983706,72383076947075470731692782081,

%U 7200266529428094485775774835670652,998383804974887102441600687728515247999,186701261436825568741051032736345268517903734

%N a(n) = U(3*n, n), where U(n, x) is the Chebyshev polynomial of the second kind.

%C In general, for k>=1, U(k*n, n) ~ 2^(k*n) * n^(k*n).

%H Seiichi Manyama, <a href="/A349074/b349074.txt">Table of n, a(n) for n = 0..136</a>

%H Eric Weisstein's World of Mathematics, <a href="https://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))^(3*n+1) - (n - sqrt(n^2-1))^(3*n+1)) / (2*sqrt(n^2-1)).

%F a(n) ~ 2^(3*n) * n^(3*n).

%t Table[ChebyshevU[3*n, n], {n, 0, 13}]

%o (PARI) a(n) = polchebyshev(3*n, 2, n); \\ _Michel Marcus_, Nov 07 2021

%o (Python)

%o from sympy import chebyshevu

%o def A349074(n): return chebyshevu(3*n,n) # _Chai Wah Wu_, Nov 08 2023

%Y Cf. A323118, A349070, A349073, A349076.

%K nonn,changed

%O 0,2

%A _Vaclav Kotesovec_, Nov 07 2021