Problem

Source: Chinese Mathematical Olympiad 2000 Problem 2

Tags: algebra unsolved, algebra



A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\ge 3$, \[a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).\] Find a closed-form expression for $f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1$.