function b = bernoulli_number ( n )

%% BERNOULLI_NUMBER computes the value of the Bernoulli numbers B(0) through B(N).
%
%  Discussion:
%
%    The Bernoulli numbers are rational.
%
%    If we define the sum of the M-th powers of the first N integers as:
%
%      SIGMA(M,N) = sum ( 0 <= I <= N ) I**M
%
%    and let C(I,J) be the combinatorial coefficient:
%
%      C(I,J) = I! / ( ( I - J )! * J! )
%
%    then the Bernoulli numbers B(J) satisfy:
%
%      SIGMA(M,N) = 1/(M+1) * sum ( 0 <= J <= M ) C(M+1,J) B(J) * (N+1)**(M+1-J)
%
%  First values:
%
%   B0  1                   =         1.00000000000
%   B1 -1/2                 =        -0.50000000000
%   B2  1/6                 =         1.66666666666
%   B3  0                   =         0
%   B4 -1/30                =        -0.03333333333
%   B5  0                   =         0
%   B6  1/42                =         0.02380952380
%   B7  0                   =         0
%   B8 -1/30                =        -0.03333333333
%   B9  0                   =         0
%  B10  5/66                =         0.07575757575
%  B11  0                   =         0
%  B12 -691/2730            =        -0.25311355311
%  B13  0                   =         0
%  B14  7/6                 =         1.16666666666
%  B15  0                   =         0
%  B16 -3617/510            =        -7.09215686274
%  B17  0                   =         0
%  B18  43867/798           =        54.97117794486
%  B19  0                   =         0
%  B20 -174611/330          =      -529.12424242424
%  B21  0                   =         0
%  B22  854,513/138         =      6192.123
%  B23  0                   =         0
%  B24 -236364091/2730      =    -86580.257
%  B25  0                   =         0
%  B26  8553103/6           =   1425517.16666
%  B27  0                   =         0
%  B28 -23749461029/870     = -27298231.0678
%  B29  0                   =         0
%  B30  8615841276005/14322 = 601580873.901
%
%  Recursion:
%
%    With C(N+1,K) denoting the standard binomial coefficient,
%
%    B(0) = 1.0
%    B(N) = - ( sum ( 0 <= K < N ) C(N+1,K) * B(K) ) / C(N+1,N)
%
%  Warning:
%
%    This recursion, which is used in this routine, rapidly results
%    in significant errors.
%
%  Special Values:
%
%    Except for B(1), all Bernoulli numbers of odd index are 0.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    22 July 2004
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer N, the order of the highest Bernoulli number to compute.
%
%    Output, real B(1:N+1), B(I+1) contains the I-th Bernoulli number.
%
  if ( n < 0 )
    b = [];
    return
  end

  b(1) = 1.0;

  if ( n < 1 )
    return
  end

  b(2) = -0.5;

  ido = 0;
 
  for i = 2 : n

    row = comb_row ( ido, i+1 );
    ido = 1;
 
    if ( mod ( i, 2 ) == 1 )
 
      b(i+1) = 0.0;
 
    else
 
      b_sum = 0.0;
      for j = 0 : i-1
        b_sum = b_sum + b(j+1) * row(j+1);
      end
 
      b(i+1) = -b_sum / row(i+1);
 
    end

  end

  return
end