Calculate the sum $\sum_{k=0}^{n-1}x^k \binom{n-1}{k} \dfrac{1}{(n-k)!}$.

Question

I want to calculate this sum: $\sum_{k=0}^{n-1}x^k \binom{n-1}{k} \dfrac{1}{(n-k)!}$. I tried to use some differentation techniques, but they didn't work. Could you help me with this?

Answer 1

Closed-form expression probably doesn't exist, but you can get a pretty good upper bound by using $\binom{n}{k} \leq \frac{n^k}{k!}$. After a bit of algebra you'll get an expression like $$ \frac{1}{n!n} \sum_{k=0}^{n-1} (n-k)\binom{n}{k} s^k $$ for some $s$.