One day Eric Temple Bell (1883–1960) was looking at the power series for the double exponential function, exp(exp(x)) and noticed a similarity to the power series for exp(x). You can find his account in [1]. He would have calculated the series by hand, but we have the advantage of software like Mathematica.
We can get the first five terms of the series, centered at 0, with the command
Series[Exp[Exp[x]], {x, 0, 5}]
This give us
If you pull out the factor of e from each term, and change the denominators to match those in the power series for exp(x) you get
with integers in all the numerators. It’s not obvious a priori that these numbers should even be integers, but they are,
Bell called the sequence numerators the exponential numbers: 1, 1, 2, 5, 15, 52, … The sequence is now known as the Bell numbers despite Bell’s modesty. Bell wasn’t the first to study this sequence of numbers, but he developed their properties more fully.
Applications
Bell numbers come up a lot in applications, which is why Bell wasn’t the first to notice them. (He may have been the first to come to them via their exponential generating function.) For example, the nth Bell number Bn is the number of ways to partition a set of n labeled items. Bn is also the nth moment of a Poisson random variable with λ = 1.
Bell’s triangle
There is a construction of Bell numbers analogous to Pascal’s triangle. Charles Sanders Peirce discovered what we now call Bell’s triangle fifty years before Bell discovered the Bell numbers.
To create Bell’s triangle, start with a row containing only 1.
The first number in each successive row is set to the last number in the previous row.
Then fill in the rest of the row by adding the number to the left and the number directly above
1
1 2
2 3 5
5 7 10 15
15 20 27 37 52
…
The numbers in the first column are the Bell numbers.
Related posts
[1] E. T. Bell. Exponential Numbers. The American Mathematical Monthly, Vol. 41, No. 7 (Aug. – Sep., 1934), pp. 411–419.