Summation of Products of n Numbers taken m at a time with Repetitions/Examples/Order 3

Theorem

 * $\displaystyle \sum_{i \mathop = 0}^n \sum_{j \mathop = 0}^i \sum_{k \mathop = 0}^j a_i a_j a_k = \dfrac { {S_1}^3} 6 + \dfrac {S_1 S_2} 2 + \dfrac {S_3} 3$

where:
 * $\displaystyle S_r := \sum_{i \mathop = 0}^n {a_i}^r$