summation - How is Faulhaber's formula derived?

I have been wanting to understand how to find the sum of this series.

$$1^p + 2^p + 3^p +{\dots} + n^p$$

I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural numbers.

I can prove the formulas for

$$\begin{align}
\sum_{1}^{n} k^2 &= \frac{n(n+1)(2n+1)}{6}\\
\sum_{1}^{n} k^3 &= \frac{n^2(n+1)^2}{4}\\

\sum_{1}^{n} k^4 &= \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} \\
\end{align}$$

With mathematical induction. But, beyond that even proofs with mathematical induction are difficult.

I'm interested in learning the theory and the proof behind Faulhaber's formula. What is the knowledge required to understand this proof ?