What is the simplest way to get Bernoulli numbers?

The simplest way to calculate them, using very few fancy tools, is the following recursive definition:

$$B_n=1-\sum_{k=0}^{n-1}{n\choose k}\frac{B_k}{n-k+1}$$ in other words $$B_n=1-{n\choose 0}\frac{B_0}{n-0+1}-{n\choose 1}\frac{B_1}{n-1+1}-\cdots -{n\choose n-1}\frac{B_{n-1}}{n-(n-1)+1}$$

Here, ${a\choose b}$ denotes a binomial coefficient. So, we begin with $B_0=1, B_1=\frac{1}{2}$, and we can calculate $B_2$ using the above recursive definition. That is, $B_2=1-{2\choose 0}\frac{B_0}{3}-{2\choose 1}\frac{B_1}{2}=1-\frac{1}{3}-2\frac{\frac{1}{2}}{2}=\frac{1}{6}$.

Now, with $B_2$ in hand, we can calculate $B_3$. And so on.