I have a nice looking recurrence relation but i cant figure out the closed form. Any help or tips are much appreciated.
$$ s_{i,k}(n) = \sum_{l=k}^{n} \dfrac{1}{l}s_{i-1,l}(n) $$ With the base condition that $s_{1,k} = \dfrac{1}{n}$
I already tried splitting the sums to get something going with the harmonic function but it doesn't paid out.
There is no hope of a closed form.
The values of $s_{1,k}$ are the suffix sums of the $n$ first terms of the ordinary harmonic series (to a factor $1/n$), which does not simplify.
And the values of the $s_{i,k}, i>2$ are even harder.
I guess that you will have asymptotically $s_{i,1}(n)=\dfrac{\log^{i-1}(n)}{ni!}$ (by successive integrations).