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) = Tn(λ).
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