Let $D:f\mapsto f'$. Then $$A(f)_x=f(x)+f(x)+f'(x)+1/2f''(x)+\cdots=(I+e^D)(f)_x.$$
Then $A^{-1}=(I+e^D)^{-1}$. 
Suppose that $f$ is a polynomial of degree $20$. Then our preferred computer calculates the  Taylor's development of $\dfrac{1}{e^x+1}$ until $O(x^{21})$.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/YYoPW.png

and we are done.

For example, $f(x)=x^3$; $A^{-1}(f)=(1/2I-1/4D+1/48D^3)(f)$, that is,
 $1/2x^3-3/4x^2+1/8$.

EDIT. In fact, $\dfrac{1}{e^x+1}=1/2(1-\tanh(x/2))$. Then the coefficients of its Taylor' development are functions of Bernouilli's numbers cf.

 https://en.wikipedia.org/wiki/Bernoulli_number

Then there exists a recurrence linking the successive coefficients.