function cx = cardan ( n, x, s, cx )

%% CARDAN evaluates the Cardan polynomials.
%
%  First terms:
%
%     N  C(N,S,X)
%
%     0  2
%     1  X
%     2  X**2  -  2 S
%     3  X**3  -  3 S X
%     4  X**4  -  4 S X**2 +  2 S**2
%     5  X**5  -  5 S X**3 +  5 S**2 X
%     6  X**6  -  6 S X**4 +  9 S**2 X**2 -  2 S**3
%     7  X**7  -  7 S X**5 + 14 S**2 X**3 -  7 S**3 X
%     8  X**8  -  8 S X**6 + 20 S**2 X**4 - 16 S**3 X**2 +  2 S**4
%     9  X**9  -  9 S X**7 + 27 S**2 X**5 - 30 S**3 X**3 +  9 S**4 X
%    10  X**10 - 10 S X**8 + 35 S**2 X**6 - 50 S**3 X**4 + 25 S**4 X**2 -  2 S**5
%    11  X**11 - 11 S X**9 + 44 S**2 X**7 - 77 S**3 X**5 + 55 S**4 X**3 - 11 S**5 X
%
%  Recursion:
%
%    Writing the N-th polynomial in terms of its coefficients:
%
%      C(N,S,X) = sum ( 0 <= I <= N ) D(N,I) * S**(N-I)/2 * X**I
%
%    then
%
%    D(0,0) = 1
%
%    D(1,1) = 1
%    D(1,0) = 0
%
%    D(N,N) = 1
%    D(N,K) = D(N-1,K-1) - D(N-2,K)
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    22 July 2004
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Thomas Osler,
%    Cardan Polynomials and the Reduction of Radicals,
%    Mathematics Magazine, 
%    Volume 74, Number 1, February 2001, pages 26-32.
%
%  Parameters:
%
%    Input, integer N, the highest polynomial to compute.
%
%    Input, real X, the point at which the polynomials are to be computed.
%
%    Input, real S, the value of the parameter, which must be positive.
%
%    Output, real CX(1:N+1), the values of the Cardan polynomials at X.
%
  s2 = sqrt ( s );
  x2 = 0.5 * x / s2;

  cx = cheby_t_polynomial ( n, x2 );

  fact = 1.0;

  for i = 0 : n
    cx(i+1) = 2.0 * fact * cx(i+1);
    fact = fact * s2;
  end
 
  return
end