Touchard polynomial

Studied by Jacques Touchard.

Touchard polynomial (plural Touchard polynomials)

1. (mathematics) Any member of a polynomial sequence of binomial type defined by ${\displaystyle T_{n}(x)=\sum _{k=0}^{n}S(n,k)x^{k}=\sum _{k=0}^{n}\left\{{n \atop k}\right\}x^{k},}$ where ${\displaystyle S(n,k)=\left\{{n \atop k}\right\}}$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.