Touchard polynomial

Etymology
Studied by.

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