recurrence formula for Bernoulli numbers

This article establishes a well-known recurrence formula for the Bernoulli numbers.

The Bernoulli polynomials $b_{r}(x),r\\geq 1$ can be written explicitly as

b_{r}(x)=\\sum_{k=1}^{r}\\binom{r}{k}B_{r-k}x^{k}+B_{r}

(see this article (http://planetmath.org/CoefficientsOfBernoulliPolynomials)).

For $r\\geq 2$ , we have

0=\\int_{0}^{1}b_{r-1}(x)dx=\\frac{1}{r}b_{r}(x)\\big{\\lvert}_{0}^{1}=\\frac{1}{r}%(b_{r}(1)-b_{r}(0))

and thus

B_{r}=b_{r}(0)=b_{r}(1)=\\sum_{k=1}^{r}\\binom{r}{k}B_{r-k}+B_{r}

It follows that (still when $r\\geq 2$ )

\\sum_{k=1}^{r}\\binom{r}{k}B_{r-k}=0

so that

\\binom{r}{1}B_{r-1}=-\\sum_{k=2}^{r}\\binom{r}{k}B_{r-k}

Replacing $r$ by $r+1$ and simplifying, we see that for $r\\geq 1$ ,

B_{r}=\\frac{-1}{r+1}\\sum_{k=2}^{r+1}\\binom{r+1}{k}B_{r+1-k}=\\frac{-1}{r+1}\\sum%_{k=1}^{r}\\binom{r+1}{k+1}B_{r-k}