Before I get started, let me say for complete disclosure this question came up while I was solving a problem from <https://projecteuler.net/>.


I've been trying to find a non-recursive representation of the following function- 
$$ S(n) = \sum\_{i=1}^{n} \frac{S(n-i)-1}{i!} $$


And I have not been able to find a method that works.  

Can anyone help by showing me a method (or an article or anything similar) to solve this type of recurrence relation?


Edit- as pointed out in the comments, the initial value is S(0)=0


Thanks in advance :)



How to solve a complex recursive relation
With $S(0)=0$ one has
$$
S(n)=-\sum\_{k=1}^n\frac{F\_k}{k!}
$$
where $F\_k$ are the [Fubini numbers](http://oeis.org/A000670) (also known as [ordered Bell numbers](https://en.wikipedia.org/wiki/Ordered_Bell_number)). The proof is contained in my comments above, given that the exponential generating function for these numbers is
$$
\sum\_{n=0}^\infty\frac{F\_n}{n!}x^n=\frac1{2-e^x}.
$$
There are many different expressions for $F\_n$ at the links to OEIS and Wikipedia that I gave, they might be used to obtain some other alternative expressions for $S(n)$; it is difficult to say which ones are more explicit and which ones less.


