coupon collector problem for different number of copies of each coupon type

Question

I would like to pose a question on a variation on the classical coupon collector's problem: coupon type $i$ is to be collected $k_i$ times. What is the expected stopping time or the expected number of trials needed to have collected all the sought after $(k_1,k_2,...,k_n)$ coupons?

We can compute this with recursion. But is there a better, more direct approach? What I am particularly interested to see is a martingale method.

Answer 1

The method in http://www.jstor.org/stable/2308930 looks like it can be adapted to this setting. It is not based on martingale arguments, but does give exact expressions.