Stating that for a sequence $\{a_n\}$ its generating function is $f(x)=\sum_{n=0}^\infty a_n x^n$. I am interested in finding out its relationship with generating function of the sequence $\{b_n=1/a_n\}$.
Tried writing $\{b_n=1/a_n=-1/(1-(a_n+1))=-\sum_{k=0}^\infty (a_n+1)^k\}$ and some infinite sums including binomial numbers and power of the original sequence. So question could "reduce" to get all generating functions from integer powers of the original sequence, so from all these k indexed set of sequences $\{c_{nk}=a_n^k\}$