How would you write a system of reccurence relations and initial conditions, based on which it would be possible to evaluate the value of the expression without repeatedly evaluating the same intermediate values for this sum $$\sum_{k=0}^\infty \frac{1+x+\dots+x^k}{k!+(k+1)!}$$
I defined \begin{align} S_0&=1/2, \\ S_k&=S_{k-1} + \frac{1+x+\dots+x^k}{k!k(k+1)!}, \quad k>0. \end{align} I'm not sure what I should do next. I mean I think I should define $a_k$ as maybe $\dfrac{1+x+\dots+x^k}{k!k(k+1)!}$ and express it through $a_{k-1}$ but... What do you think?