Asymptotic of Stirlings numbers of the first kind I am trying to find some asymptotic expression for the unsigned stirling numbers of the first kind. 
Lets denote them by $|s(n,k)|$, and suppose that $k$ is fixed.
So far I have tried using the fact that $$|s(n,k)| = \sum\{ i_1i_2\ldots i_{n-k} : 1 \leq i_1 < i_2 < \ldots < i_{n-k} \leq n-1 \}$$
And trying to start from $$ \frac{1}{(n-k)!}\left( \sum_{i=1}^{n-1}i \right)^{n-k} = \frac{1}{(n-k)!} {n \choose 2}^{n-k}$$
Which contains all products of the form $\{ i_1i_2\ldots i_{n-k} : i_j \in [0,n-1] \}$ divided by that factor of $\frac{1}{(n-k)!}$, so I was trying to sepparate the products of distinct terms from the rest, and trying to see that the rest should be "small". But I got stuck from here.
So, can somebody give me any tips, or help me with this approach, or should I look for asymptotics by using the recurrence relation, the generating function or maybe some other technique?
Thanks,
 A: For small $k$ you can get the asymptotic behavior in $n$ from explicit formulas for 
$\left[ \matrix{n\\k}\right]$, in terms of Harmonic numbers $H_m \equiv \sum_{0<i\leq m} m^{-1}$ and generalized harmonic numbers $H_m^{(p)} \equiv \sum_{0<i\leq m} m^{-p}$:
$$
\left[ \matrix{n\\1}\right] = (n-1)! \\
\left[ \matrix{n\\2}\right] = (n-1)! H_{n-1}\\
\left[ \matrix{n\\3}\right] = \frac12 (n-1)! \left( (H_{n-1})^2 - H_{n-1}^{(2)}\right)
$$
These are derived from the basic recursion relation. 
Since the harmonic and generalized harmonic numbers are well-studied, this immediately gives the expansions for those cases:
$$
\left[ \matrix{n\\2}\right] = (n-1)! \left( \log n + \gamma - \frac1{2n} - \frac1{12n^2} + O(n^{-3}) \right) \\
$$
$$
\left[ \matrix{n\\3}\right] = (n-1)! \left( \frac12 (\log n)2 + \gamma \log n - \frac1{12} (\pi^2 - 6\gamma^2) - \frac{\log n}{2n} + \frac{1-\gamma}{2n} - \frac{\log n}{12n^2} + \frac{9-2\gamma}{24n^2}+ O(n^{-3}\log n) \right) 
$$
I could not get the $k=4$ case in closed form, but working with the recursion I was able to find that it starts with
$$
\left[ \matrix{n\\3}\right] = (n-1)! \left( \frac16 (\log n)^3 +  O((\log n)^2) \right) 
$$
These results agree with those of the paper cited by @Vaclav Kotesovec.
For $k$ comparable to a fraction of $n$, the problem is much harder, although Moser has a paper in a 1958 London Journal of Mathematics which might deal with it; I can't get my hands on that paper.
