Bernoulli numbers

Except for this one difference, the first and second Bernoulli numbers agree.

For example, he is credited with the discovery of Bernoulli numbers.

However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd values.

As a result, the Bernoulli numbers have the distinction of being the subject of the first computer program.

This recurrence relation may be derived from that for the Bernoulli numbers.

What he did was come up with a formula that explains and simplifies the "Bernoulli numbers."

The particular choice provides an explicit representation of the Bernoulli numbers, since .

However, both simple and high-end algorithms for computing Bernoulli numbers exist.

The Bernoulli numbers can be regarded from four main viewpoints:

For the even positive integers, one has the relationship to the Bernoulli numbers:

The Bernoulli numbers appear in the book in a discussion of the exponential series.

Bernoulli numbers as values of a sequence of certain polynomials.

Bernoulli numbers are also frequently used in other kinds of asymptotic expansions.

It can be shown that has a nice expression in terms of the Bernoulli numbers whenever s is a positive even integer.

This integral is well-known and can be expressed in terms of Bernoulli numbers:

An alternate convention defines the Bernoulli numbers as .

Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers.

Bernoulli numbers as values of the Riemann zeta function.

Bernoulli numbers can be expressed through the Euler numbers and vice versa.

He also devised a method of calculating B based on previous Bernoulli numbers.

Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index.

Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers.

Here the are the Bernoulli numbers and is the Glaisher-Kinkelin constant.

Why do the odd Bernoulli numbers vanish?

The are the usual Bernoulli numbers.