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A193678
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Discriminant of Chebyshev C-polynomials.
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11
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1, 8, 108, 2048, 50000, 1492992, 52706752, 2147483648, 99179645184, 5120000000000, 292159150705664, 18260173718028288, 1240576436601868288, 91029559914971267072, 7174453500000000000000, 604462909807314587353088, 54214017802982966177103872
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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MAPLE
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seq(discrim(2*orthopoly[T](n, x/2), x), n = 1..50); # Robert Israel, Aug 04 2015
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MATHEMATICA
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t=Table[NumberFieldDiscriminant[2^(1/m)], {m, 1, 20}] (* signed version *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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