  \displaystyle  \sum_{k=1}^{n-1} \sum_{j=k+1}^{n} k \cdot j  =   \frac{1}{2} \cdot   \left (  \sum_{k=1}^{n}  k^{3}  -  \sum_{k=1}^{n}  k^{2}    \right )  =   \frac{1}{2} \cdot   \left (  \sum_{k=1}^{n}  k^{2}  \left ( k-1 \right )   \right )  