I need to find the generating function for $a_n$ where:
$$a_{2k}=\dfrac{(-1)^k}{(2k)!}$$
$$a_{2k+1}=\dfrac{1}{(2k+1)}$$
$$k \geq 0$$
The solution is:
$$f(x)=\cos(x)+\operatorname{atanh}(x)$$
So far I've solved problems where everything began with a recurrence relation that I had to multiply with something to get sums (and substitute with $f(x)$).
I'm not sure how to even express $a_n$ here. I'd appreciate any help.