A127606 - OEIS

%I #29 Nov 08 2023 01:48:04

%S 1,4,176,79808,372713728,17931360207872,8887976555024756736,

%T 45390122553039546330628096,2388340820825093234015277927170048,

%U 1294826675280341699389150405743029631844352

%N a(n) = 2^(2*n*n) * Product_{1<=i,j<=n} (cos(i*Pi/(2*n+1))^2 + sin(j*Pi/(2*n+1))^2).

%C a(n)/4^n is an integer. - _Seiichi Manyama_, Dec 31 2020

%H Seiichi Manyama, <a href="/A127606/b127606.txt">Table of n, a(n) for n = 0..40</a>

%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) = 2^n * sqrt(Resultant(T_{2*n+1}(i*x/2), U_{2*n}(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). - _Seiichi Manyama_, Jan 09 2021

%F a(n) ~ 2^(1/8) * exp(G*(2*n + 1)^2/Pi) / (1 + sqrt(2))^(n + 1/2), where G is Catalan's constant A006752. - _Vaclav Kotesovec_, Mar 18 2023

%p for n from 0 to 12 do a[n]:=2^(2*n*n)*product(product(cos(i*Pi/(2*n+1))^2+ sin(j*Pi/(2*n+1))^2,j=1..n),i=1..n) od: seq(round(evalf(a[n],300)),n=0..12);

%t Table[2^(2*n^2) * Product[Product[Cos[i*Pi/(2*n + 1)]^2 + Sin[j*Pi/(2*n + 1)]^2, {i, 1, n}], {j, 1, n}], {n, 0, 15}] // Round (* _Vaclav Kotesovec_, Mar 18 2023 *)

%o (PARI) default(realprecision, 120);

%o {a(n) = round(prod(i=1, n, prod(j=1, n, 4*cos(i*Pi/(2*n+1))^2+4*sin(j*Pi/(2*n+1))^2)))} \\ _Seiichi Manyama_, Dec 31 2020

%o (PARI) {a(n) = sqrtint(4^n*polresultant(polchebyshev(2*n+1, 1, I*x/2), polchebyshev(2*n, 2, x/2)))} \\ _Seiichi Manyama_, Jan 09 2021

%o (Python)

%o from math import isqrt

%o from sympy.abc import x

%o from sympy import resultant, chebyshevt, chebyshevu, I

%o def A127606(n): return isqrt(resultant(chebyshevt((n<<1)+1,I*x/2),chebyshevu(n<<1,x/2)))<<n if n else 1 # _Chai Wah Wu_, Nov 07 2023

%Y Cf. A004003, A127605.

%K nonn

%O 0,2

%A _Miklos Kristof_, Apr 03 2007