Graham, Knuth and Patashnik in their book Concrete Mathematics called for development of a general theory of the solutions of recurrences defined by $$\left|{ n\atop k}\right|=(αn+βk+γ)\left|{n-1\atop k}\right|+(α' n+β' k+γ')\left|{n-1\atop k-1}\right|+I_{n=k=0}$$ for $0\le k\le n$ and six parameters $α,β,γ,α'β',γ'$. Since then, a number of authors investigated various properties of the solutions of these recurrences. In this note we consider a probabilistic aspect, namely 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 $α'=0$ and the remaining five parameters are non--negative.