Where do the Bernoulli numbers come from? I'm trying to get an idea of how the definition of the Bernoulli numbers was settled on. I found this explanation very clear and intuitive, relating the Bernoulli numbers to the sums of a fixed power. 
The one part I can't quite see is how Bernoulli (or anyone) would have gone from the empirical evidence to the general pattern. From what I can see, the following facts seem clear to me:


*

*The leading coefficient is always $1/(p+1)$. 

*The second coefficient is always $-1/2$

*The third coefficient is always $p/12$

*The fourth coefficient is always $0$.


Seeing the $1/(p+1)$ and $p$ factors, it seems reasonable that you could guess that we have a falling factorial factor: $(p+1)p(p-1)...(p-k)/(p+1)$. And dividing out by this factor leaves us (miraculously) with an invariant quantity.
So I guess my question is why weren't these quantities used as the Bernoulli numbers? What kind of reasoning led Bernoulli to divide by an additional factor of $k!$ ?
 A: If you scroll down a bit in your linked page, you'll see a "modern definition" given as coefficients appearing in the power series expansion: 
$$\frac{z}{e^z-1} = \sum_{k=0}^\infty B_k \frac{z^k}{k!}$$
where the Bernoulli numbers $B_k$ may be identified with $k^{th}$ derivatives of the function $G(z)=z/(e^z-1)$ at $z=0$.
This is a sufficiently common construction as to have a name: $G(z)$ is said to be the generating function of the Bernoulli numbers $B_k$ (by virtue of the above power series expression).  Note that the factorial denominators appear by virtue of having these repeated derivatives in the numerator for any power series expansion.
The term "Bernoulli numbers" was naturally not used or defined by Jakob Bernoulli, whom we commemorate with this naming.  The interested Reader might enjoy the note Bernoulli numbers by John C. Baez (2003) and its jocular but insightful opening:

They are called the Bernoulli numbers because they were first studied by Johann Faulhaber in a book published in 1631, and mathematical discoveries are never named after the people who made them. For example, the ‘Cech compactification’ was invented by Tychonoff, while ‘Tychonoff’s Theorem’ is due to Cech.

We do honor Johann Faulhaber by naming after him Faulhaber's formula, by which the sums of $p$ powers of $1,\ldots,n$ are expressed as a $(p+1)$-degree polynomial in $n$:
$$ \begin{align*} \sum_{k=1}^n k^p &= 1^p+2^p+\ldots+n^p \\
   &= \frac{1}{p+1} \sum_{j=0}^p (-1)^j \binom{p+1}{j} B_j \; n^{p+1-j} 
   \end{align*} $$
This generality is beyond what Faulhaber or Bernoulli could have expressed. Indeed we can find in Faulhaber's book, Academia Algebrae - Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden (1631), a copy of which (in German) may be viewed online, merely the first seventeen instances of those polynomials.  
Bernoulli recites a smaller number of instances, but he describes a procedure to generate them recursively.  An English translation of his Latin book Ars Conjectandi was done by Edith Dudley Sylla, titled The Art of Conjecturing, together with Letter to a Friend on Sets in Court Tennis (out of print but available in many university libraries).  The formula instances from the image above appear on page 215 (see Part Two) of this English translation, following the discussion of Faulhaber's previous work and the procedure for generating them.
For more information see the Wikipedia articles on generating functions and on Bernoulli numbers, as well as this older Math.SE Question.
Note that the name Bernoulli numbers was also used for $B_k^*$, a related but different sequence in the older literature, as explained by the Wolfram MathWorld article, although you have to scroll pretty far down the page (to equation (52)) to see the details.  These include a factor $(2k)!$ rather than $k!$, so it is an alternative "normalization" for coefficients.
