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Closed form expression for f_n for sequence a_n

Source: Chinese Mathematical Olympiad 2000 Problem 2

August 18, 2013
algebra unsolvedalgebra

Problem Statement

A sequence (an)(a_n) is defined recursively by a1=0,a2=1a_1=0, a_2=1 and for n≥3n\ge 3, an=12nan−1+12n(n−1)an−2+(−1)n(1−n2).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 fn=an+2(n1)an−1+3(n2)an−2+…+(n−1)(nn−2)a2+n(nn−1)a1f_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.
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