  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.
