Bell number

Partitions of sets can be arranged in a partial order, showing that each partition of a set of size n "uses" one of the partitions of a set of size n-1.
The triangular array whose right-hand diagonal sequence consists of Bell numbers

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.

1, 3, 13, 12, 781, 39, 137257, 24, 39, 2343, 28531167061, 156, 25239592216021, 411771, 10153, 48, 51702516367896047761, 39, 109912203092239643840221, 9372, 1784341, 85593501183, 949112181811268728834319677753, 312, 3905, 75718776648063, 117, 1647084, 91703076898614683377208150526107718802981, 30459, 568972471024107865287021434301977158534824481, 96, 370905171793, 155107549103688143283, 107197717, 156, ... (sequence in the OEIS)

An application of Cauchy's integral formula to the exponential generating function yields the complex integral representation

2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 (sequence in the OEIS)

corresponding to the indices 2, 3, 7, 13, 42 and 55 (sequence in the OEIS).