Dobinski’s formula entails the following infinite series expression for the nth Bell number :
In this post we’ll work through a proof of Dobinski’s formula. We’ll need four formulas:
- The Maclaurin series for : .
- The formula for converting normal powers to falling factorial powers: where is a Stirling number of the second kind.
- The formula relating Bell numbers and Stirling numbers of the second kind: . (This follows directly from the combinatorial definitions of the Bell numbers and the Stirling numbers of the second kind.)
- The representation of falling factorial powers as factorials: , valid when and .
Here we go… Starting with the infinite series expression convert to falling factorial powers to get
(I made a similar argument in this Math.SE question.)