This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193678 #33 Feb 15 2025 20:26:23
%S A193678 1,8,108,2048,50000,1492992,52706752,2147483648,99179645184,
%T A193678 5120000000000,292159150705664,18260173718028288,1240576436601868288,
%U A193678 91029559914971267072,7174453500000000000000,604462909807314587353088,54214017802982966177103872
%N A193678 Discriminant of Chebyshev C-polynomials.
%C A193678 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).
%C A193678 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.
%C A193678 One could add a(0)=0 for the discriminant of C(0,x)=2.
%C A193678 Except for sign, a(n) is the field discriminant of 2^(1/n); see the Mathematica program. - _Clark Kimberling_, Aug 03 2015
%D A193678 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.
%H A193678 Robert Israel, <a href="/A193678/b193678.txt">Table of n, a(n) for n = 1..320</a>
%H A193678 Sinan Deveci, <a href="https://arxiv.org/abs/2211.10751">On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function</a>, arXiv:2211.10751 [math.NT], 2022.
%F A193678 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.
%F A193678 a(n) = (2^(n-1))*n^n, n>=1.
%F A193678 a(n) = A000079(n-1)*A000312(n). - _Omar E. Pol_, Aug 27 2011
%e A193678 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).
%p A193678 seq(discrim(2*orthopoly[T](n,x/2), x), n = 1..50); # _Robert Israel_, Aug 04 2015
%t A193678 t=Table[NumberFieldDiscriminant[2^(1/m)], {m, 1, 20}] (* signed version *)
%t A193678 Abs[t] (* _Clark Kimberling_, Aug 03 2015 *)
%t A193678 Table[(2^(n - 1)) n^n, {n, 20}] (* _Vincenzo Librandi_, Aug 04 2015 *)
%o A193678 (Magma) [(2^(n-1))*n^n: n in [1..20]]; // _Vincenzo Librandi_, Aug 04 2015
%Y A193678 Cf. A127670.
%K A193678 nonn,easy
%O A193678 1,2
%A A193678 _Wolfdieter Lang_, Aug 07 2011