Proving of an E.G.F

Question

We wish to prove that the exponential generating function of the sequence defined by the recurrence relations $$a_0 = 1, a_1 = 1$$ $$a_n = na_{n-1} + (n-1)a_{n-2}$$ is given by $\frac{e^{-x} }{(1-x)^2}$

We came across this series in the context of some research and wish to compute the e.g.f via the method of Taylor series.

Answer 1

I try to prove the e.g.f using the Taylor series as it requires, is this correct?

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