Touchard polynomials

The Touchard polynomials comprise a polynomial sequence of binomial type defined by

where S(n, k) is a Stirling number of the second kind, i.e., it is the number of partitions of a set of size n into k disjoint non-empty subsets. (The second notation above, with { braces }, was introduced by Donald Knuth.)

If X is a random variable with a Poisson distribution with expected value λ, then its nth moment is E(Xⁿ) = T_n(λ).

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