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 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 AUTHOR Wolfdieter Lang, Aug 07 2011 STATUS approved

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Last modified August 6 07:05 EDT 2024. Contains 374960 sequences. (Running on oeis4.)