Two experiments with different probabilities of success run independently $n_1$ and $n_2$ times. I'm modeling the number of successes of each experiment as two independent binomial random variables: $X\sim\mathcal B(n_1, p_1)$ and $Y\sim\mathcal B(n_2, p_2)$.
I would like to know $\Pr[X + Y = k]$ for a constant $k$, i.e., the probability that the sum of the successes of the two experiments in $n_1+n_2$ trials is $k$.
Is there an expression in terms of $n_i$, $p_i$, and $k$ for such probability?