Numerators of Maclaurin series coefficients

Question

I have noticed that often the Maclaurin series of notable functions have rational coefficients whose denominators are relatively easy to understand, while the numerators are intractable. Two examples I can think of are the Bernoulli numbers (coefficients of $\displaystyle \frac{x}{e^x-1}$ ) and the coefficients of $\sec(x)$ (both of interest in combinatorics). I have tabulated the coefficients of $\sec(x)$ as far as my computer power allowed and I have found that the numerators are often large primes and apparently always square free. I would be interested in knowing if the fact that numerators of Maclaurin series are often hard to understand while the denominators are easier is discussed or studied somewhere. Is there some underlying principle to explain it? I searched the archives and the web but did not find anything.