``
        The Recurrence operator annihilating a_r(n):=The number of ways

 of making change of n cents only using pennies, nickels, dimes, and quarter\
    s

          where the person has r pockets the money can come from, is:

                       2      3      4                    5        6        7
(-41 r - n) - r N - r N  - r N  - r N  + (-5 - n - 47 r) N  - 2 r N  - 2 r N

            8        9         10        11        12        13        14
     - 2 r N  - 2 r N  - 22 r N   - 2 r N   - 2 r N   - 2 r N   - 2 r N   -

          15        16        17        18        19         20        21
    22 r N   - 2 r N   - 2 r N   - 2 r N   - 2 r N   - 22 r N   - 2 r N   - 2 r

     22        23        24                   25      26      27      28
    N   - 2 r N   - 2 r N   + (25 + n - 6 r) N   - r N   - r N   - r N   - r

     29             30
    N   + (30 + n) N

                           This took, 1.572, seconds