Description (from Rosetta Code)
| 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:
| 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.
Generate the first 10 closed-form expressions, starting with p = 0.
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: :