The n=0 term is 0, so you can work with the sum for n>0.
For n>0, one n in the n^3 cancels with the n!.
Pull an x out of the infinite sum. Now you've essentially got the same sum, but
with the index variable as x-1. Rename this to y.
The top now has a (y+1)^2. Expand this to y^2+2y+1, split the infinite sum into 3 infinite sums, and use this trick again.