The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A193678 Discriminant of Chebyshev C-polynomials. 11
 1, 8, 108, 2048, 50000, 1492992, 52706752, 2147483648, 99179645184, 5120000000000, 292159150705664, 18260173718028288, 1240576436601868288, 91029559914971267072, 7174453500000000000000, 604462909807314587353088, 54214017802982966177103872 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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)/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 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..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 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^(n-1))*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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 28 05:56 EST 2022. Contains 358407 sequences. (Running on oeis4.)