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.

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