I am looking for a a family of probability mass functions $f_n$ with the following recurrence: $$ f_{n}(k)=qf_{n-1}(k)+p\sum_{i=1}^{k-1}f_{n-1}(i)f_{n-1}(k-i). $$ In addition, we have $q=1-p$ and $ p,q \in [0,1]$ and $$ f_{0}(k)=\begin{cases} 1; & k=1,\\ 0; & \mbox{otherwise.} \end{cases} $$
In terms of generating function $F_{n}(x)=\sum_{i=0}^{\infty}f_{n}(i)\,x^{i}$ this simplifies to $$ F_{n}(x)=qF_{n-1}(x)+pF_{n-1}(x)^{2}. $$
With Sage, I am able to calculate functions up to $n\leq 8$, but cannot guess any nice solution so far. For illustration, the first few members of $F_n$ are $$ \begin{eqnarray*} F_{1}(x) & = & px^{2}-{\left(p-1\right)}x,\\ F_{2}(x) & = & p^{3}x^{4}-2\,{\left(p^{3}-p^{2}\right)}x^{3}+{\left(p^{3}-3\,p^{2}+2\,p\right)}x^{2}+{\left(p^{2}-2\,p+1\right)}x,\\ F_{3}(x) & = & p^{7}x^{8}-4\,{\left(p^{7}-p^{6}\right)}x^{7}+2\,{\left(3\,p^{7}-7\,p^{6}+4\,p^{5}\right)}x^{6} -2 \,{\left(2\,p^{7}-9\,p^{6} +12\,p^{5}-5\,p^{4}\right)}x^{5} +{\left(p^{7}-10\,p^{6}+25\,p^{5}-25\,p^{4}+9\,p^{3}\right)}x^{4} +2\,{\left(p^{6}-5\,p^{5}+10\,p^{4}-9\,p^{3}+3\,p^{2}\right)}x^{3} +{\left(p^{5}-5\,p^{4}+10\,p^{3}-9\,p^{2}+3\,p\right)}x^{2} -{\left(p^{3}-3\,p^{2}+3\,p-1\right)}x. \end{eqnarray*} $$