In this presentation by Mathologer you can find the following slide:
relating the sum of the $n$ first $k$ powers of integers, i.e. $S_k$ and the Faulhaber matrix diagonals (which correspond to the columns highlighted on the image above. Here is the corresponding Faulhaber matrix:
S 0 1 0 0 0 0 0 0
S 1 1/2 1/2 0 0 0 0 0
S 2 1/6 1/2 1/3 0 0 0 0
S 3 0 1/4 1/2 1/4 0 0 0
S 4 -1/30 0 1/3 1/2 1/5 0 0
S 5 0 -1/12 0 5/12 1/2 1/6 0
S 6 1/42 0 -1/6 0 1/2 1/2 1/7
The patterns along the first four columns are well explained: The harmonic sequence, $1/2$'s, $k/12$, $0$'s, and then the following statement:
Jacob Bernoulli discovered something amazing: The numbers in the different columns always appear to depend in a simple way on numbers at the top of these columns.
Unfortunately he doesn't share the simple dependence of the fifth (or higher) column: $-1/30, -1/12, -1/6, - 7/24, -7/15, -7/10,\dots.$
or formulate what type of general dependence this is.
Let $a_{k,m}$ denote the coefficient of the $k$th column of $S_m$ (corresponding to the monomial $x^{m-k+2}$).
The pattern for the first column ($k=1$) is $$a_{1,m} = \left(\frac{m}{m+1}\right) a_{1,m-1}.\quad (m\ge 1)$$
The pattern for the second column ($k=2$) is $$a_{2,m} = \left(\frac{m}{m}\right) a_{2,m-1}.\quad (m\ge 2)$$
The pattern for the third column ($k=3$) is $$a_{3,m} = \left(\frac{m}{m-1}\right) a_{3,m-1}.\quad (m\ge 3)$$
You can see that the pattern for any $k$ is $$a_{k,m} = \left(\frac{m}{m-k+2}\right) a_{k,m-1},\quad (m\ge k)$$
where the denominator is easily remembered as the power of $x$ attached to the coefficient $a_{k,m}$. In other words, to get the coefficient of $x^r$ in $S_m$, take the coefficient of $x^{r-1}$ in $S_{m-1}$ and multiply it by $\frac{m}{r}$.