#This is a holonomic representation for the random variable" number of color-changes when you draw
#balls form an urn with m White balls and n Black Balls

[- (x + 1) (x - 1) m (m + 1)

      2                2                                  /
    (m  x - 2 m n x + n  x + 3 m x - 3 n x + 2 n + 2 x)  /  ((m + n + 2)
                                                        /

                  2                2                           3  3      2    3
    (m + n + 1) (m  x - 2 m n x + n  x + m x - n x + 2 n)) + (m  x  - 3 m  n x

            2  3    3  3      2  3          3      2  3      3        2
     + 3 m n  x  - n  x  + 3 m  x  - 6 m n x  + 3 n  x  - 2 m  x + 4 m  n x

            2            2        3      2  2        3      2
     - 2 m n  x + 4 m n x  + 2 m x  - 4 n  x  - 2 n x  - 6 m  x + 8 m n x

          2          2                            /
     - 2 n  x + 4 n x  - 4 m n - 4 m x - 4 n) M  /  ((m + n + 2)
                                                /

      2                2                          2
    (m  x - 2 m n x + n  x + m x - n x + 2 n)) + M , - (x + 1) (x - 1) n

              2                2                                  /
    (n + 1) (m  x - 2 m n x + n  x - 3 m x + 3 n x + 2 m + 2 x)  /  (
                                                                /

                              2                2                           3  3
    (m + n + 2) (m + n + 1) (m  x - 2 m n x + n  x - m x + n x + 2 m)) - (m  x

          2    3        2  3    3  3      2  3          3      2  3      2
     - 3 m  n x  + 3 m n  x  - n  x  - 3 m  x  + 6 m n x  - 3 n  x  + 2 m  n x

          2  2        2            2        3      3          3      2
     + 4 m  x  - 4 m n  x - 4 m n x  + 2 m x  + 2 n  x - 2 n x  + 2 m  x

                      2      2                              /
     - 8 m n x - 4 m x  + 6 n  x + 4 m n + 4 n x + 4 m) N  /  ((m + n + 2)
                                                          /

      2                2                          2
    (m  x - 2 m n x + n  x - m x + n x + 2 m)) + N ]

                          This took, 88.078, seconds.