The problem:
There are n different kinds of coupons, and we need to get every kind of coupon at least once to win. Each time we get new coupons we get them in a batch of k coupons where k ≤ n (the same coupon can appear multiple times in the batch of k coupons).
Let Xk,n be a random variable that denotes the number of batches we need to buy to collect all the n different coupons.
I already know and understand how to calculate the expected value when k=1 (that is the classic coupon collector problem), but have difficulty calculating E[Xk,n] for k > 1.
If it helps, I've also made some simulations and estimations, and came up with this table:
Let $k=2$. Let $E(m)$ be the expected number of pairs needed to complete $n$ coupons when you already have $m$ coupons.
The chance of ending with $m+1$ coupons comes from three cases: First new, second old;First old, second new;Both the same new coupon. $$E(m)=1+\frac{(n-m)(n-m-1)}{n^2}E(m+2)+\frac{2m(n-m)+(n-m)}{n^2}E(m+1)+\frac{m^2}{n^2}E(m)$$ Then you build the matrix to get numerical answers; or solve the algebra to get exact answers.
For example, $E(n-1)=1+\frac{(n-1)^2}{n^2}E(n-1)=\frac{n^2}{2n-1}$