Identify The Series!

Calculus Level 5

$\dfrac { n! }{ { a }_{ n-1 } } ={ a }_{ n }$

Consider a recurrence relation above for $n = 1,2,3,\ldots$ and $a_0 = 1$ . If the value of the series $\displaystyle \sum_{n=0}^\infty \dfrac1{a_{2n} }$ can be expressed as $e^{\alpha /\beta}$ , where $\alpha$ and $\beta$ are coprime positive integers, find $\alpha+\beta$ .

Clarification: $e \approx 2.71828$ denotes the Euler's number.