<s>[INST] How to get formula for sums of powers? Assuming I have Bernoulli numbers:
$B = [\frac{1}{1},\frac{1}{2},\frac{1}{6},\frac{0}{1},-\frac{1}{30}, \frac{0}{1}, \frac{1}{42}, ...]$
How can I get the coefficients of the sums of powers formulas?
For example the sum of squares is $(1/3)n^3 + (1/2)n^2 + 1/6n$
 [/INST] A vey-well rememberable scheme is perhaps the following.      
Use the vector of Bernoulli-numbers in the first row of a little scheme and multiply through each column (the result per column is in the last row of the table):
$$ \begin{bmatrix}\frac{1}{1}&\frac{1}{2}&\frac{1}{6}&\frac{0}{1}&-\frac{1}{30}& \frac{0}{1}& \frac{1}{42}& ... \\
*x^4 & *x^3&*x^2&*x& & & &  \\ 
/4 & /3& /2&/1& & & &  \\ 
*1 & *3& *3&*1& & & &  \\ \hline
1/4x^4 & 1/2x^3& 1/4x^2& 0x& & & & 
\end{bmatrix}$$
Sum of cubes is $1/4x^4 + 1/2x^3+ 1/4x^2$
$$ \begin{bmatrix}\frac{1}{1}&\frac{1}{2}&\frac{1}{6}&\frac{0}{1}&-\frac{1}{30}& \frac{0}{1}& \frac{1}{42}& ... \\
*x^5 & *x^4&*x^3&*x^2&*x & & &  \\ 
/5 & /4& /3&/2&/1 & & &  \\ 
*1 & *4& *6&*4&*1 & & &  \\ \hline
1/5x^5 & 1/2x^4& 1/3x^3& 0x^2&-1/30x 
\end{bmatrix}$$
sum-of-$4$'th powers is $ 1/5x^5 + 1/2x^4 + 1/3x^3  -1/30x $
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