Here are the first five rows of the triangle constructed by these rules:
The Bell numbers appear on both the left and right sides of the triangle.
A different summation formula represents each Bell number as a sum of Stirling numbers of the second kind
Spivey (2008) has given a formula that combines both of these summations:
In this formula, the summation in the middle is the general form used to define the exponential generating function for any sequence of numbers, and the formula on the right is the result of performing the summation in the specific case of the Bell numbers.
An application of Cauchy's integral formula to the exponential generating function yields the complex integral representation
corresponding to the indices 2, 3, 7, 13, 42 and 55 (sequence in the OEIS).