The Bernoulli polynomials are the sequence $\{ b_r(x) \} _{r=0}^{\infty}$ of polynomials defined on $[0,1]$ by the conditions: \begin{eqnarray*} b_0(x) & = & 1, \\ b'_r(x) & = & r b_{r-1}(x), r \geq 1, \\ \int_0^1 b_r(x)dx & = & 0, r \geq 1 \end{eqnarray*} These assumptions imply the identity $$ \sum_{r=0}^{\infty} b_r(x) \frac{y^r}{r!} = \frac{ye^{xy}}{e^y-1} $$ allowing us to calculate the $b_r$ We have

\begin{eqnarray*} b_0(x) & = & 1 \\ b_1(x) & = & x-\frac{1}{2} \\ b_2(x) & = & x^2 - x + \frac{1}{6} \\ b_3(x) & = & x^3 - \frac{3}{2}x^2 + \frac{1}{2}x \\ b_4(x) & = & x^4 - 2x^3 + x^2 - \frac{1}{30} \\ \vdots & & \end{eqnarray*}