## Notation for growth $a_n \le c (n!)^\epsilon$

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?

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Terminology or notation? For the latter, $\log(1+|a_n|)=o(n\log n)$. For the former, I do not know. – Didier Piau Jun 14 2011 at 17:42
It might help to take logarithms of both sides before looking for a name. Gerhard "Ask Me About System Design" Paseman, 2011.06.14 – Gerhard Paseman Jun 14 2011 at 17:44
@Didier: thanks, I was mainly interested in terminology, so something like $1+|a_n|$ is of $n\log(n)$ growth does not seem to be less clumsy... – Stefan Waldmann Jun 14 2011 at 17:59
@Gerhard: thanks for the comment, maybe I don't get it but I don't see how the estimate of Didier suggest a better name for the type of growth. – Stefan Waldmann Jun 14 2011 at 18:00

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) .