The distribution of the sum of two independant random variables is (IIRC) the convolution product of the distributions. From that, it's rather simple to compute it :

use List::Util qw/sum/;
my @a; # holds the known distribution
sub conv {
 my ($n) = @_;
 return sum map {
  ($_ < @a) ? ((($n - $_) < @a) ? $a[$_] * $a[$n - $_]
                                : 0)
            : 0
 } 0 .. $n;
}
[download]

If you want the distribution of the sum of n random variables, you can adapt this code to accept two distributions and recursively compute the n-th convolution product of @a.