Suppose we have an unbiased multinomial random variable with $n$ trials and $k$ categories. So $X_1, \dots, X_k$ sum to $n$ and have jointly a multinomial distribution with probability $1/k, \dots, 1/k$.
For any $x_1, \dots, x_k$, we have $$ p(x_1, \dots, x_k) = \binom{n}{x_1, \dots, x_k} k^{-n} $$
I would like to estimate $$ S = \sum_{x_1, \dots, x_k} p(x_1, \dots, x_k)^2 $$ that is, the probability that I get two repeated identical draws from the multinomial.
For $k$ fixed, we apparently have $s = (c_k + o(1)) n^{-(k-1)/2}$. This makes sense: each coordinate varies on the order of $\sqrt{n}$, and so the probability of a repetition in a single coordinate is about $n^{-1/2}$. The first $k-1$ coordinates are independent-ish.
Can one compute this constant term $c_k$?
Any references or keywords to search for would be much appreciated! Thanks!