Touchard Polynomial Definition

Any member of a polynomial sequence of binomial type defined by T_n(x)=\sum_{k=0}^n S(n,k)x^k=\sum_{k=0}^n \left\{ {n \atop k} \right\}x^k, where S(n,k)=\left\{ {n \atop k} \right\}is a Stirling number of the second kind , i.e. the number of partitions of a set of size n into k disjoint non-empty subsets.