We consider sequences of polynomials that satisfy differential-difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete mathematics. It is, therefore, of interest to understand the properties of such polynomials and their probabilistic consequences. We identify a class of polynomial recurrences that lead to a normal law with the expected value and the variance proportional to $n/\log n$ and $n/\log^2n$, respectively. Examples include Stirling number of the second kind and other polynomials concerning set partitions as well as polynomials related to Whitney numbers of Dowling lattices.