Summation of Products of n Numbers taken m at a time with Repetitions/Examples/Order 4

Theorem

 * $\displaystyle \sum_{1 \mathop \le j_1 \mathop \le j_2 \mathop \le j_3 \mathop \le j_4 \mathop \le n} x_{j_1} x_{j_2} x_{j_3} x_{j_4} = \dfrac { {S_1}^4} {24} + \dfrac { {S_1}^2 S_2} 4 + \dfrac { {S_2}^2} 8 + \dfrac {S_3} 3 + \dfrac {S_4} 4$

where:
 * $\displaystyle S_r := \sum_{k \mathop = 1}^n {x_k}^r$

Proof
From Summation to n of Summation to Index:


 * $\displaystyle \sum_{1 \mathop \le j_1 \mathop \le \cdots \mathop \le j_m \mathop \le n} x_{j_1} \cdots x_{j_m} = \sum_{\substack {k_1, k_2, \ldots, k_m \mathop \ge 0 \\ k_1 \mathop + 2 k_2 \mathop + \cdots \mathop + m k_m \mathop = m} } \dfrac { {S_1}^{k_1} } {1^{k_1} k_1 !} \dfrac { {S_2}^{k_2} } {2^{k_2} k_2 !} \cdots \dfrac { {S_m}^{k_m} } {m^{k_m} k_m !}$

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

Setting $m = 4$:

We need to find all sets of $k_1, k_2, k_3, k_4 \in \Z_{\ge 0}$ such that:


 * $k_1 + 2 k_2 + 3 k_3 + 4 k_4 = 4$

Thus $\left({k_1, k_2, k_3, k_4}\right)$ can be:


 * $\left({4, 0, 0, 0}\right)$
 * $\left({2, 1, 0, 0}\right)$
 * $\left({0, 2, 0, 0}\right)$
 * $\left({1, 0, 1, 0}\right)$
 * $\left({0, 0, 0, 1}\right)$

Hence: