Transforming Exponential to Ordinary Generating Functions

Question

I am looking for a particular ordinary generating function, if it exists for the Associated Stirling Numbers of the second kind $$b(1;n,j)=b(n,j)=\sum_{k=0}^j(-1)^k\binom{n}{k}S(n-k,j-k)$$
Where $S(n,k)$ are the Stirling Numbers of the Second Kind. I am motivated to take up the search again after reading this page. It mentions that "whenever possible" you can convert the function using a Laplace Transform. I am interested in finding out about this technique and was hoping to see papers or books that deal specifically with this kind of problem.