Suppose $X_1,\dots,X_n$ are jointly distributed random variables such that the random $n$-tuple $(X_1,\dots,X_n)$ is uniformly distributed on the set of $n$-tuples of nonnegative integers summing to $k$. Clearly each $X_i$ has expected value $k/n$. Are there nice formulas for the expected values of $X_i^2$, $X_i X_j$, etc.?