Bernoulli's formula

In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers

as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers B_j.

The formula says

For example, the case p = 1 is

Faulhaber himself did not know the formula in this form, but only computed the first seventeen polynomials; the general form was established with the discovery of the Bernoulli numbers (see History section below). The derivation of Faulhaber's formula is available in The Book of Numbers by John Horton Conway and Richard K. Guy.

There is also a similar (but somehow simpler) expression: using the idea of telescoping and the binomial theorem, one gets Pascal's identity:

This in particular yields the examples below, e.g., take k = 1 to get the first example.

Let

denote the sum under consideration for integer ${\displaystyle p\geq 0.}$

Define the following exponential generating function with (initially) indeterminate ${\displaystyle z}$

...
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