Let \(a_0 = a_1 =1\) and \(a_n = n a_{n-1} + (n-1) a_{n-2}\) for \(n \geq 2\). Find the value of

\[

\sum_{n=0}^{\infty} (-1)^n \frac{n!}{a_n a_{n+1}}.

\]

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Let \(a_0 = a_1 =1\) and \(a_n = n a_{n-1} + (n-1) a_{n-2}\) for \(n \geq 2\). Find the value of

\[

\sum_{n=0}^{\infty} (-1)^n \frac{n!}{a_n a_{n+1}}.

\]

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