Probability of irregular 10-sided die

#1
I've got an odd question here:
I've got a 10-sided die with sides labeled as such: -1, 0, 0, 0, 0, 1, 1, 1, 1, 2
Thus, the odds for a single die look like this:
Odds of -1: 0.1
Odds of 0: 0.4
Odds of +1: 0.4
Odds of +2: 0.1

I'm trying to find the probabilities of rolling every possible outcome starting with rolling 1 die (easy enough) to 10 dice. I've attempted to do this programmatically, but I'm having a hard time figuring out the best way to do it.

Any suggestions?