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 $$ b_r(x) = \sum_{k=1}^r \binom{r}{k}B_{r-k}x^k + B_ $$ (see this article).

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

It follows that (still when $r\geq 2$ ) $$ \sum_{k=1}^r \binom{r}{k}B_{r-k}= $$ 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 $r\geq 1$ , $$ B_r = \frac{-1}{r+1}\sum_{k=2}^{r+1}\binom{r+1}{k}B_{r+1-k} = \frac{-1}{r+1}\sum_{k=1}^r\binom{r+1}{k+1}B_{r-k $$