What does the generating function $x/(1 - e^{-x})$ count?

Two people have pointed it out already, but somehow I can't resist: your formal power series is precisely the defining power series of the Bernoulli numbers:

http://en.wikipedia.org/wiki/Bernoulli_number#Generating_function

Accordingly, they are far from Egyptian: as came up recently in response to the question

When does the zeta function take on integer values?

the odd-numbered terms (except the first) are all zero, whereas the even-numbered terms alternate in sign and grow rapidly in absolute value, so only finitely many are reciprocals of integers.

I find it curious that you are looking at this sequence from such a sophisticated perspective and didn't know its classical roots. I feel like there should be a lesson here, but I don't know exactly what it is. Here's a possibility: every young mathematician should learn some elementary number theory regardless of their primary interests. Comments?

Here's another way to get at the answer. You think you have a sequence of rationals that may be familiar:

1, 1/2,1/12,0,-1/720,...

The denominators seem more interesting than the numerator, so maybe the "right" sequence is:

1,2,12,1,720,...

You go to Sloane's Encyclopedia and enter the sequence, to no avail. You could now try superseeker, which looks at many transformations of the sequence, but for this few terms that will return too many hits. Let's try the one transformation you mentioned, and look at the exponential generating function, whose coefficients have denominators:

1, 2, 6, 1, 30, ...

Sloane's immediately identifies that sequence as the denominators of Bernoulli numbers, giving not only the generating function you started with but many other interesting factoids and references.

I am adding the following remark because it may be of some interest to the number theorists who recognized the Bernoulli numbers to know that the relationship with Lie theory explained in the question has number-theoretic substance: namely, in his article on the thrice-punctured sphere, Deligne uses the Lie algebra point of view on Bernoulli numbers described in the question (together with other ingredients, of course, and applied to a specific Lie algebra) to derive Euler's formula for the values of $\zeta(2n)$.