Thinking of some questions of homotopical algebra for operads, I ended up with a following question, perhaps someone will recognize something here:
Let $\{a_n\}_{n\ge 2}$ be a sequence of nonnegative integers. Let's say that this sequence has the property S if for the series $$ f(t)=t+\sum_{n\ge 2} \frac{a_n}{n!}t^n, $$ the series $$ f^{<-1>}(f(t)-t) $$ has non-negative coefficients. Here $f^{<-1>}$ is the reversion (compositional inverse) of $f$. For example, the sequence $a_n=(n-1)!$ has the property S and the sequence $a_n=n!$ does not, these are easy exercises. It seems more difficult (but true) that the sequence $a_n=n^{n-1}$ has the property S.
My question is whether someone saw something of that sort before, or has some relevant references. In particular, I am interested to know if can prove in a very indirect way that the sequence $\{a_n\}$ for which $f^{<-1>}(t)=t-t^2+\frac{t^5}{120}$ has the property S, but I am wondering if the same can be established by a direct combinatorial argument of some sort.
