A193678 - OEIS

%I #30 Mar 02 2023 08:02:12

%S 1,8,108,2048,50000,1492992,52706752,2147483648,99179645184,

%T 5120000000000,292159150705664,18260173718028288,1240576436601868288,

%U 91029559914971267072,7174453500000000000000,604462909807314587353088,54214017802982966177103872

%N Discriminant of Chebyshev C-polynomials.

%C The array of coefficients of the (monic) Chebyshev C-polynomials is found under A127672 (where they are called, in analogy to the S-polynomials, R-polynomials).

%C See A127670 for the formula in terms of the square of a Vandermonde determinant, where now the zeros are xn[j]:=2*cos(Pi*(2*j+1)/n), j=0,..,n-1.

%C One could add a(0)=0 for the discriminant of C(0,x)=2.

%C Except for sign, a(n) is the field discriminant of 2^(1/n); see the Mathematica program. - _Clark Kimberling_, Aug 03 2015

%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219 for T and U polynomials.

%H Robert Israel, <a href="/A193678/b193678.txt">Table of n, a(n) for n = 1..320</a>

%H Sinan Deveci, <a href="https://arxiv.org/abs/2211.10751">On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function</a>, arXiv:2211.10751 [math.NT], 2022.

%F a(n) = (Det(Vn(xn[1],..,xn[n]))^2, with the n x n Vandermonde matrix Vn and the zeros xn[j],j=0..n-1, given above in a comment.

%F a(n) = (2^(n-1))*n^n, n>=1.

%F a(n) = A000079(n-1)*A000312(n). - _Omar E. Pol_, Aug 27 2011

%e n=3: The zeros are [sqrt(3),0,-sqrt(3)]. The Vn(xn[1],..,xn[n]) matrix is [[1,1,1],[sqrt(3),0,-sqrt(3)],[3,0,3]]. The squared determinant is 108 = a(3).

%p seq(discrim(2*orthopoly[T](n,x/2), x), n = 1..50); # _Robert Israel_, Aug 04 2015

%t t=Table[NumberFieldDiscriminant[2^(1/m)], {m, 1, 20}] (* signed version *)

%t Abs[t] (* _Clark Kimberling_, Aug 03 2015 *)

%t Table[(2^(n - 1)) n^n, {n, 20}] (* _Vincenzo Librandi_, Aug 04 2015 *)

%o (Magma) [(2^(n-1))*n^n: n in [1..20]]; // _Vincenzo Librandi_, Aug 04 2015

%Y Cf. A127670.

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Aug 07 2011