A334124 a(n) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{2*n}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

1, 3, 71, 17753, 46069729, 1234496016491, 341133743251787719, 971684488369988888850993, 28523907708086181923163934073729, 8628515016553040037389969912341438652243, 26895841132028233579514694272575933932911355677831

Offset: 0

Author

Seiichi Manyama, Apr 15 2020

Keywords

• nonn

Links

• Wikipedia, Chebyshev polynomials
• Wikipedia, Resultant

Crossrefs

Main diagonal of A103997.

Cf. A004003, A334088.

Programs

Mathematica

Table[2^n * Sqrt[Resultant[ChebyshevU[2*n, x/2], ChebyshevT[2*n, I*x/2], x]], {n, 0, 12}] (* Vaclav Kotesovec, Apr 16 2020 *)

PARI

{a(n) = sqrtint(4^n*polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(2*n, 1, I*x/2)))}

Python

from math import isqrt
from sympy import resultant, chebyshevt, chebyshevu, I
from sympy.abc import x
def A334124(n): return isqrt(resultant(chebyshevu(n<<1,x/2),chebyshevt(n<<1,I*x/2))*(1<<(n<<1))) if n else 1 # Chai Wah Wu, Nov 07 2023

Formula

a(n) = A103997(n,n).

a(n) ~ 2^(1/4) * exp(2*G*n*(2*n+1)/Pi) / (1 + sqrt(2))^n, where G is Catalan's constant A006752. - Vaclav Kotesovec, Apr 16 2020, updated Jan 03 2021