First 10 Bernoulli numbers:

Bernoulli polynomials:

BernoulliB threads element-wise over lists:

Plot Bernoulli polynomials:

TraditionalForm formatting:

Find sums of powers using BernoulliB (Faulhaber's formula):

Compare with direct summation:

Set up an Euler–Maclaurin integration formula:

Use it for :

Compare with the exact summation result:

Plot roots of Bernoulli polynomials in the complex plane:

Show the approach of Bernoulli numbers to a limiting form:

The denominator of Bernoulli numbers is given by the von Staudt–Clausen formula:

Compute Bernoulli numbers in modular arithmetic modulo a prime:

Find BernoulliB numbers from their generating function:

Find BernoulliB polynomials from their generating function:

BernoulliB can be represented as a DifferenceRoot:

Algorithmically produced results are frequently expressed using Zeta instead of BernoulliB:

When entered in the traditional form, is not automatically interpreted as a Bernoulli number:

Going from Bernoulli numbers to Bernoulli polynomials with umbral calculus:

The 20000 Bernoulli number can be computed in under a second:

Define a Hankel matrix whose entries are the Bernoulli numbers:

Its determinant can be expressed in terms of the Barnes G-function: