In combinatorics, the Bell number Bn gives the number of ways of
partitioning a set of size n into non-empty subsets.
Dobinski's formula gives the nth Bell number, Bn = 1/e * sigma
(k=0,infinity) k^n/k! There is also a variation of Dobinski
which gives:
Bn = sigma (k=1,n) k^n/k! * sigma (j=0,n-k) (-1)^j/j!
The Bell Triangle can also be used, by taking advantage of the recurrence
relation Bn+1 = sigma (k=0,n) Bk ( n ) where ( a ) is a binomial coefficient.
(
k ) ( b )
There are other functions that can be used to generate Bell numbers, such
as Comtet's formula, exponential polynomials, and Stirling Transformations.
As a point of interest, there are only 6 Bell numbers less than 1000 that
are also prime, and they are B2,3,7,13,42,55.
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