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 $(1/3)n^3 + (1/2)n^2 + 1/6n$

Stiring Numbers of the Second Kind is also good for building sum of powers formula. 
$x^n = \sum _{k=0} ^n S(n,k) (x)_k$
Example, for sum of x^2 formula, "integrate" the falling factorial form.
$$x^2 = (1)(x)_1 + (1)(x)_2$$
$$\sum_{k=0}^{n-1} x^2 
= \frac{(n)_2}{2}+\frac{(n)_3}{3}
= \frac{n^3}{3} - \frac{n^2}{2} + \frac{n}{6}$$
