I need some guidance with the following problem.

I have a sequence $L\_0,L\_1,\ldots$ whose ordinary generating series satisfies $$L(x) = \sum\_{n=0}^{\infty} L\_n \frac{x^n}{n!} = \frac{1}{2-e^x}.$$

Now the exercise I am doing asks to find a linear differential equation that is satisfied by $L(x)$ and use it to obtain a recurrence for $L\_n.$

I am very rusty and don't even know how to proceed. I can see that, for example $$x L'(x) - L(x) = \sum\_{n=0}^{\infty} (n-1)L\_n \frac{x^n}{n!}$$ but I have no idea how to obtain anything useful from this direction.

So my question is :

how does one obtain a linear recurrence satisfied by $L\_n$ this case?