SpecialEquation

Problem - 1804
Let the sequence {$a_n$} satisfy $a_0=0$, $a_1=1$, $a_{n+2} = (n+3)a_{n+1} -(n+2)a_n$. Find whether the following equation is solvable in rational numbers:$$\sum_{i=1}^n\frac{x^i}{a_i-a_{i-1}}=-1\qquad\qquad(n \ge 2)$$

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