Graham, Knuth and Patashnik in their book Concrete Mathematics called for development of a general theory of the solutions of recurrences defined by
\[\left|{ n\atop k}\right|=(\alpha n+\beta k+\gamma)\left|{n-1\atop k}\right|+(\alpha' n+\beta' k+\gamma')\left|{n-1\atop k-1}\right|+I_{n=k=0}\]
for $0\le k\le n$ and six parameters $\alpha,\beta,\gamma,\alpha', \beta',\gamma'$. Since then, a number of authors investigated various properties of the solutions of these recurrences, often under restrictions on the range of the parameters. In this talk we consider the limiting distributions of sequences of integer valued random variables naturally associated with the solutions of such recurrences. We will give a complete description of the limiting behavior when $\alpha'=0$ and the remaining five parameters are non--negative.