I was trying to solve a recurrence using Exponential Generating Function but I can't figure out what to do with one of the terms. The recurrence is:
$$RQ_0 = 0$$ $$RQ_1 = 0$$ $$RQ_n = n + \frac{2}{2^n}\sum_{k=0}^n \binom{n}{k}RQ_k$$
We then multiply both sides by $2^n$ and try to find the EGF, but I don't know how to handle the term $\sum_{n\ge2}n2^n\frac{z^n}{n!}$
The teacher want us to prove $Â(z) = z\sum_{j\ge0}(e^z - e^{(1 - 2^{-j})z})$, where $Â(z)$ is the EGF for $RQ$.