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

http://mathoverflow.net/questions/7586/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?