This is just a stupid question about a good terminology. I'm interested in sequences $a\_n$ with a growth that can be bounded by an arbitrarily small positive power of $n!$, i.e. for every $\epsilon > 0$ there should be a constant $c$ with $|a\_n| \le c (n!)^{\epsilon}$. I wanted to call this growth "sub-factorial" but then I learned that there is already a "sub-factorial" $!n$. Any good suggestions?



Notation for growth $a_n \le c (n!)^\epsilon$
In the need of a name, I would also go for "sub-factorial growth". In this case I see no danger of confusion; "sub-linear" and "sub-exponential growth" are already used in analogous meaning (even if not completely standard). Also note that in your condition you can replace $n!$ with $n^n$ (which unfortunately didn't give me better hints) . 


