An intuitive meaning of Bernoulli numbers

[ddddd28]

Recently, I was intrigued by the summations of finite powers and therefore by the formula which generalizes the summations. "Faulhaber's formula".

However, I didn't find an intuitive simple meaning of "bernoulli numbers", only meaning by their applications, which, of course, I can't understand them. I am looking for an explanation similar to how pascal numbers are understood, for example.

 

[fresh_42]

I find already this formula satisfactory. Or
$$
\frac{x}{e^x-1} = \sum_{k=0}^\infty B_k\, \frac{x^k}{k!} \textrm{ or } \tan x = \sum_{k=1}^\infty (-1)^k \, \frac{2^{2k}(1-2^{2k})}{(2k)!}\,B_{2k}x^{2k-1}
$$
and many more of similar type. I don't think this answers your question as they are basically of the same kind as Faulhaber's formula above. So I assume there is no answer as simple as the Pascal triangle. The Wikipedia page on Bernoulli's numbers says, that Bernoulli's original approach was by the powers of natural numbers, too. So the question which is really interesting is, what makes them so "universal" that they appear in so many different contexts as up to the Riemannian zeta-function. And this question probably breaks down to the definition via the exponential function above and the importance of power series in general.

 

[ddddd28]

So the question which is really interesting is, what makes them so "universal" that they appear in so many different contexts as up to the Riemannian zeta-function.

I think you are right, I would like to understand why there are so universal, and in which cases do I know that I should use them?