Touchard polynomials
The Touchard polynomials, studied by Jacques Touchard (1956), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by T n ( x ) = ∑ k = 0 n S ( n , k ) x k = ∑ k = 0 n { n k } x k , {\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 S ( n , k ) = { n k } {\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. The first few Touchard polynomials are T 1 ( x ) = x , {\displaystyle T_{1}(x)=x,} T 2 ( x ) = x 2 + x , {\displaystyle T_{2}(x)=x^{2}+x,} T 3 ( x ) = x 3 + 3 x 2 + x , {\displaystyle T_{3}(x)=x^{3}+3x^{2}+x,} T 4 ( x ) = x 4 + 6 x 3 + 7 x 2 + x , {\displaystyle T_{4}(x)=x^{4}+6x^{3}+7x^{2}+x,} T 5 ( x ) = x 5 + 10 x 4 + 25 x 3 + 15 x 2 + x .