The Bernoulli numbers satisfy the recursive relationship . (Here, is the *Iverson bracket*, where evaluates to 1 if *P* is true and 0 if *P* is false.) The relationship can be used to calculate the Bernoulli numbers fairly easily. This post gives a proof of this relationship.

The exponential generating function for the Bernoulli numbers is given by . The exponential generating function for the sequence consisting of 0 followed by an infinite number of 1’s is given by . Multiplying these together, we obtain, thanks to the multiplication principle for exponential generating functions (see, for example, *Concrete Mathematics* [1], p. 365)

But has generating function . Thus we have . Replacing with *n*, we have the formula we were trying to prove:

.

(As a side note, the *Concrete Mathematics* reference, p. 365, derives the exponential generating function for the Bernoulli numbers from the recurrence relation.)

**References**

1. Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. *Concrete Mathematics*, 2nd edition. Addison-Wesley, 1994.