# Faulhaber

This page is a solution to the tasks Faulhaber's triangle and Faulhaber's formula in the Rosetta Code, written in Fōrmulæ.

## Contents

### Description (from Rosetta Code)

#### Faulhaber's triangle

 Named after Johann Faulhaber, the rows of Faulhaber's triangle are the coefficients of polynomials that represent sums of integer powers, which are extracted from Faulhaber's formula: where is the nth-Bernoulli number. The first 5 rows of Faulhaber's triangle, are: ``` 1 1/2 1/2 1/6 1/2 1/3 0 1/4 1/2 1/4 -1/30 0 1/3 1/2 1/5 ``` Using the third row of the triangle, we have: Task show the first 10 rows of Faulhaber's triangle. using the 18th row of Faulhaber's triangle, compute the sum: (extra credit). See also

#### Faulhaber'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. Task Generate the first 10 closed-form expressions, starting with p = 0. Related tasks See also The Wikipedia entry: Faulhaber's formula. The Wikipedia entry: Bernoulli numbers. The Wikipedia entry: binomial coefficients.

### Solution

The following function creates the Faulhaber's coefficients up to a given number of rows, according to the paper of Mohammad Torabi Dashti:

To show the first 10 rows of Faulhaber's triangle:

In order to show the previous result as a triangle:

The following function creates the sum of the p-th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n:

Notes. The -1 index means the last element (-2 is the penultimate element, and so on). So it retrieves the last row of the triangle. |x| is the cardinality (number of elements) of x.

To generate the first 10 closed-form expressions, starting with p = 0:

Extra credit. Using the 18th row of Faulhaber's triangle, compute the sum: :