What are Bernoulli numbers?

Bernoulli numbers are a sequence of rational numbers that arise in number theory, particularly in the calculation of sums of powers of integers and in the expansion of certain functions. They are defined using the generating function ( \frac{x}{e^x - 1} = \sum_{n=0}^{\infty} B_n \frac{x^n}{n!} ), where ( B_n ) are the Bernoulli numbers. These numbers play a crucial role in various mathematical areas, including combinatorics, calculus, and the theory of special functions. They also appear in the Euler-Maclaurin formula and in the computation of the Riemann zeta function at even integers.