$S_1(n,0)=S_1(0,n)= \delta_n \; \;$ and, for $n > 0$, 

$$S_1(n,k)= \lim_{y \to 0} \sum_{j=1}^k (-1)^j \binom{k}{j} \; \frac{(-j \; y)!}{(-j \; y-n)!} \; \frac{y^{-k}}{k!}$$

$$ = \lim_{y \to 0} \sum_{j=1}^k (-1)^{n-j} \binom{k}{j} \; \frac{(j \; y - 1 + n)!}{(j \; y-1)!} \; \frac{y^{-k}}{k!} \; .$$

For a derivation, see [A class of differential operators and the Stirling numbers][1]. Note that with $y$ small enough taking the nearest integer generates $S_1$.


  [1]: http://tcjpn.wordpress.com/2015/08/23/a-class-of-differential-operators-and-the-stirling-numbers/#more-2614