A193678 Discriminant of Chebyshev C-polynomials.

1, 8, 108, 2048, 50000, 1492992, 52706752, 2147483648, 99179645184, 5120000000000, 292159150705664, 18260173718028288, 1240576436601868288, 91029559914971267072, 7174453500000000000000, 604462909807314587353088, 54214017802982966177103872

Offset: 1

Author

Wolfdieter Lang, Aug 07 2011

Keywords

• nonn
• easy

Comments

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).

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)/(2*n)), j=0,..,n-1.

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

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

Examples

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

References

• 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.

Links

• Robert Israel, Table of n, a(n) for n = 1..320
• Sinan Deveci, On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function, arXiv:2211.10751 [math.NT], 2022.

Crossrefs

Cf. A127670.

Programs

Magma

[(2^(n-1))*n^n: n in [1..20]]; // Vincenzo Librandi, Aug 04 2015

Maple

seq(discrim(2*orthopoly[T](n,x/2), x), n = 1..50); # Robert Israel, Aug 04 2015

Mathematica

t=Table[NumberFieldDiscriminant[2^(1/m)], {m, 1, 20}] (* signed version *)
Abs[t] (* Clark Kimberling, Aug 03 2015 *)
Table[(2^(n - 1)) n^n, {n, 20}] (* Vincenzo Librandi, Aug 04 2015 *)

Formula

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

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

a(n) = A000079(n-1)*A000312(n). - Omar E. Pol, Aug 27 2011