What are the coefficients of the series for: $$\frac x{e^x+1}$$ It looks similar to the Bernoulli generating function, but the $+$ sign is throwing me off.
I already found the series for its inverse. If $x=\frac y{e^y+1}$, then: $$y=\sum_{n=1}^\infty\left(\sum_{k=1}^n\binom nkk^{n-1}\right)\frac{x^n}{n!}$$ —that is, that finite series in the middle is the $n$th coefficient of the exponential generating function. (Wolfram Alpha can't reduce it to a simpler form.) (If you're bored later today, by the way, it's a nice challenge to derive that.)
They are Genocchi numbers, when you multiply by 2. In particular even Genocchi numbers are related to Bernoulli numbers:
$$G_{2k}=2(1-2^{2k})B_{2k}.$$