Finding Polynomial Equations for $1^4 + 2^4 + 3^4 + \ldots n^4$ Find a polynomial expression for
$$ 1^4 + 2^4 + 3^4 + ... + n^4 $$
I know you have to use the big theorem but I can't figure out how you would start to compute the differences. Suggestions?

There is a simpler formula without using the Bernoulli numbers,
$$(n+1)((n+1)^p-1) = \sum_{k=1}^{p} {p+1 \choose k }S_k $$
where $S_p = 1^p + 2^p + \dots + n^p$
For example, $p=1$
$$ (n+1)(n+1- 1)= {2 \choose 1} S_1 = 2S_1$$
$$ S_1 = \frac{n(n+1)}2 $$
Similarly, for $p=4$
$$ (n+1)((n+1)^4 - 1) = 5S_1 + 10S_2 + 10S_3 +5S_4$$
from where $S_4$ can be easily computed in terms of lower sums, but this representation is easier to remember.
Note this is a generalization of @Babak's answer.
