A334088 - OEIS

%I #20 Nov 08 2023 01:49:33

%S 1,1,8,676,591872,5347119376,497996601804800,477995151754478453824,

%T 4727827717838439286122217472,481856411624794348153802518369517824,

%U 506033683217425527860454091268429289861152768

%N a(n) = sqrt(Resultant(T(2*n,x/2), T(2*n,i*x/2))), where T(n,x) is a Chebyshev polynomial of the first kind and i = sqrt(-1).

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

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

%F a(n) ~ exp(4*G*n^2/Pi) / 2^(2*n - 1/4), where G is Catalan's constant A006752. - _Vaclav Kotesovec_, Apr 14 2020

%t Table[Sqrt[Resultant[ChebyshevT[2*n, x/2], ChebyshevT[2*n, I*x/2], x]], {n, 0, 12}] (* _Vaclav Kotesovec_, Apr 14 2020 *)

%o (PARI) {a(n) = sqrtint(polresultant(polchebyshev(2*n, 1, x/2), polchebyshev(2*n, 1, I*x/2)))}

%o (Python)

%o from math import isqrt

%o from sympy.abc import x

%o from sympy import resultant, chebyshevt, I

%o def A334088(n): return isqrt(resultant(chebyshevt(n<<1,x/2),chebyshevt(n<<1,I*x/2))) if n else 1 # _Chai Wah Wu_, Nov 07 2023

%Y Cf. A004003, A334089.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 14 2020