function value = euler_poly ( n, x )

%% EULER_POLY evaluates the N-th Euler polynomial at X.
%
%  First values:
%
%    E(0,X) = 1
%    E(1,X) = X - 1/2
%    E(2,X) = X**2 - X
%    E(3,X) = X**3 - 3/2 X**2 + 1/4
%    E(4,X) = X**4 - 2*X**3 + X
%    E(5,X) = X**5 - 5/2 X**4 + 5/2 X**2 - 1/2
%    E(6,X) = X**6 - 3 X**5 + 5 X**3 - 3 X
%    E(7,X) = X**7 - 7/2 X**6 + 35/4 X**4 - 21/2 X**2 + 17/8
%    E(8,X) = X**8 - 4 X**7 + 14 X**5 - 28 X**3 + 17 X
%
%  Special values:
%
%    E'(N,X) = N * E(N-1,X)
%
%    E(N,1/2) = E(N) / 2**N, where E(N) is the N-th Euler number.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    27 July 2004
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer N, the order of the Euler polynomial to
%    be evaluated.  N must be 0 or greater.
%
%    Input, real X, the value at which the polynomial is to
%    be evaluated.
%
%    Output, real VALUE, the value of E(N,X).
%
  bx1 = bernoulli_poly2 ( n+1, x );
  bx2 = bernoulli_poly2 ( n+1, 0.5 * x );

  value = 2.0  * ( bx1 - bx2 * 2.0^( n + 1 ) ) / ( n + 1 );

  return
end