Answer & Explanation
The value of is , where is the n-th Bell number.
To see this, note that
This is precisely the recursion formula that the Bell numbers follow, except that every term here is being multiplied by e.
A basic technique in real (complex) analysis is term by term differentiation of power series:
Evaluating at , one immediately has
Combining the second and third equalities, we have the answer.
Just to give a slightly different approach,