

A193678


Discriminant of Chebyshev Cpolynomials.


11



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

1,2


COMMENTS

The array of coefficients of the (monic) Chebyshev Cpolynomials is found under A127672 (where they are called, in analogy to the Spolynomials, Rpolynomials).
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,..,n1.
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


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.


FORMULA

a(n) = (Det(Vn(xn[1],..,xn[n]))^2, with the n x n Vandermonde matrix Vn and the zeros xn[j],j=0..n1, given above in a comment.
a(n)= (2^(n1))*n^n, n>=1.
a(n) = A000079(n1)*A000312(n).  Omar E. Pol, Aug 27 2011


EXAMPLE

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).


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 *)


PROG

(Magma) [(2^(n1))*n^n: n in [1..20]]; // Vincenzo Librandi, Aug 04 2015


CROSSREFS

Cf. A127670.
Sequence in context: A215129 A234571 A120975 * A265277 A336828 A184267
Adjacent sequences: A193675 A193676 A193677 * A193679 A193680 A193681


KEYWORD

nonn,easy,changed


AUTHOR

Wolfdieter Lang, Aug 07 2011


STATUS

approved



