Identify The Series!

Calculus Level 5

n!an1=an\dfrac { n! }{ { a }_{ n-1 } } ={ a }_{ n }

Consider a recurrence relation above for n=1,2,3,n = 1,2,3,\ldots and a0=1a_0 = 1. If the value of the series n=01a2n \displaystyle \sum_{n=0}^\infty \dfrac1{a_{2n} } can be expressed as eα/β e^{\alpha /\beta} , where α\alpha and β\beta are coprime positive integers, find α+β\alpha+\beta.

Clarification: e2.71828e \approx 2.71828 denotes the Euler's number.

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