Jelly, 5 4 bytes
Ḋ²ḋṖ
How?
Implements \$\sum_{i=2}^n{(i^2(i-1))}\$...
Ḋ²ḋṖ - Link: non-negative integer, n
Ḋ - dequeue (implicit range) [2,3,4,5,...,n]
² - square (vectorises) [4,9,16,25,...,n*n]
Ṗ - pop (implicit range) [1,2,3,4,...,n-1]
ḋ - dot product 4*1+9*2+16*3+25*4+...+n*n*(n-1)