Given the sequence $(a_n)_{n\geq0}$, the elements of which are recursively defined as follows:
$$a_0 := 0; a_{n+1} := n \cdot a_n + 1$$
We can define a formal power series $A(z) = \sum_{n\geq0}a_nz^n$ and after a few simplification steps we obtain: $$A(z) = a_0 + z \cdot \sum_{n\geq0}n \cdot a_n \cdot z^n + z \cdot A(z)$$
My goal is to obtain a generating function by eliminating the power series but I am not sure how to proceed. None of the "simple" power series known to me fit the pattern of the power series above.
Is there some Identity I can use or is there another trick?
Thanks for the help! :)