For fixed y1, calculate P(Y1 = y1, Y2 = 0) y referring back to the original random variables. If y1 = 0, this is just the probability that Y2 = 0, while if y1 != 0, the probability is 0. That will take care of the case Y2 = 0, and you can sum Y2 out in the usual way.

(I would be more clear but Latex is acting up, sorry )