Statistical Math Question

I'm developing an opposed contest system based on dice pools. A 6 is an exploding success that adds 1 success to the total and can be rerolled once more, yielding either a success (4 or higher) or nothing (3 or lower). A 4 or 5 is a simple success. A 2 or 3 is a simple failure. A 1 is an imploding failure, which subtracts 1 success. If there are no successes, or there are more 1s than successes, this *would* result in a negative number of successes.

Basically, I am trying to figure out the statistical distribution for X number of die in order to be able to define my success/failure range for skill contests.

EXAMPLE:

For a skill contest where one side is rolling a pool of 3 dice, the range would be between -3 and +6.

Could someone point to some existing formulae or statistical technique I could use to quickly calculate the probabilities for each possibility within this range? I want it to be as accurate and reasonably balanced as possible, but not necessarily representing true human performance metrics. My RPG is going to be remarkably Gamist, and doesn't need to accurately represent plausible human accomplishment so much as it needs to be balanced and symmetrical.

Anyway, if someone could point out how to properly calculate this for any number of dice (I'll go easy and assume that the maximum number of dice used will be somewhere around 10), it would be a big help to me.