Faulhaber

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

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: :