###### Problem 384

Consider the recurrence \(a_n=na_{n-1} +n(n-1)\text{.}\) Multiply both sides by \(\frac{x^n}{n!}\text{,}\) and sum from \(n=2\) to \(\infty\text{.}\) (Why do we sum from \(n=2\) to infinity instead of \(n=1\) or \(n=0\text{?}\)) Letting \(y = \sum_{i=0}^\infty a_ix^i\text{,}\) show that the left-hand side of the equation is \(y-a_0 -a_1x\text{.}\) Express the right hand side in terms of \(y\text{,}\) \(x\text{,}\) and \(e^x\text{.}\) Solve the resulting equation for \(y\) and use the result to get an equation for \(a_n\text{.}\) (A finite summation is acceptable in your answer for \(a_n\text{.}\))