Geodesic
A generalized recurrence for Bell numbers
Journal of integer sequences, Tome 11 (2008) no. 2
We show that the two most well-known expressions for Bell numbers, $ \varpi_n = \sum_{k=0}^n \genfrac{\{}{\}}{0pt}{}{n}{k}$ and $ \varpi_{n+1} = \sum_{k=0}^n \binom{n}{k} \varpi_k$, are both special cases of a third expression for the Bell numbers, and we give a combinatorial proof of the latter.
Classification : 11B73
Keywords: Bell number, Stirling number 
@article{JIS_2008__11_2_a1,
     author = {Spivey,  Michael Z.},
     title = {A generalized recurrence for {Bell} numbers},
     journal = {Journal of integer sequences},
     year = {2008},
     volume = {11},
     number = {2},
     zbl = {1231.11026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2008__11_2_a1/}
}
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Spivey,  Michael Z. A generalized recurrence for Bell numbers. Journal of integer sequences, Tome 11 (2008) no. 2. http://geodesic.mathdoc.fr/item/JIS_2008__11_2_a1/