Summation of Products of n Numbers taken m at a time with Repetitions

Theorem
Let there be $n$ numbers $x_1, x_2, \ldots, x_n$.

Let:
 * $h_m = \displaystyle \sum_{i \mathop \le j_1 \mathop \le \cdots \mathop \le j_m \mathop \le n} x_{j_1} \cdots x_{j_m}$

for some $m \in \Z_{>0}$.

Let $S_j = \displaystyle \sum_{k \mathop = 1}^n {x_k}^j$ for $j \in \Z_{\ge 0}$.

Then:


 * $h_m = \displaystyle \sum_{\substack {k_1, k_2, \ldots, k_m \mathop \ge 0 \\ k_1 \mathop + 2 k_2 \mathop + \cdots \mathop + m k_m \mathop = m} } \dfrac { {S_1}^{k_1} } {1^{k_1} k_1 !} \dfrac { {S_2}^{k_2} } {2^{k_2} k_2 !} \dfrac { {S_m}^{k_m} } {m^{k_m} k_m !}$

Proof
Let us set up the generating function for the sequence $\left\langle{h_m}\right\rangle$:

Thus, by definition of generating function:
 * $h_m = \displaystyle \sum_{\substack {k_1, k_2, \ldots, k_m \mathop \ge 0 \\ k_1 \mathop + 2 k_2 \mathop + \cdots \mathop + m k_m \mathop = m} } \dfrac { {S_1}^{k_1} } {1^{k_1} k_1 !} \dfrac { {S_2}^{k_2} } {2^{k_2} k_2 !} \dfrac { {S_m}^{k_m} } {m^{k_m} k_m !}$