For $1 \le i \le n$, let $X_i$ be $n$ integer-valued random variables that sum to the constant $s$. None of the $X_i$ are zero and they can have identical values. I need to find the the expected value of the product of the random variables, i.e. $E[X_1 X_2\cdots X_n]$. I have been told that there is a simple expression for the answer:
$$E[X_1X_2\cdots X_n] = s(s+1)(s+2)\cdots(s+n-1)/n(n+1)(n+2)\cdots(2n-1).$$
Example: $s = 9$, $n = 5$. The values $X_1, X_2, X_3, X_4, X_5$ can take are all possible arrangements of the following integer partitions of $9$, i.e.
By direct calculation, $E[X_1X_2X_3X_4X_5] = 715/70 = 143/14$
Using the suggested answer, $E[X_1X_2X_3X_4X_5] = 9\cdot10\cdot12\cdot13/5\cdot6\cdot7\cdot8\cdot9 = 143/14$
I'm not sure whether this is a well-known result, and I have not been able to find any references to it. I have verified that the result holds for various values of s and n, but I am not sure how to prove the result.