Problem Statement

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

B

n

{\displaystyle B_{n}}

is the nth-Bernoulli number.

The first 5 rows of Faulhaber's triangle, are:

Using the third row of the triangle, we have:

∑

k

1

n

k

2

=

1 6

n +

1 2

n

2

1 3

n

3

{\displaystyle \sum _{k=1}^{n}k^{2}={1 \over 6}n+{1 \over 2}n^{2}+{1 \over 3}n^{3}}

Let's start with the solution: