For positive integers $n,k$, define $$f(n,k):=\sum\_{i=1}^{n-1}(n-i)\binom{k}{i}.$$


What are upper and lower bounds of $f(n,k)$ by simpler terms? (e.g. finding bounds which are not a summation like this one.) How fast does $f(n,k)$ grow asymptotically in $n$ and $k$?


If we loosely bound $\binom{k}{i}\leq 2^k$, then we get $f(n,k)\leq 2^{k-1}n(n-1)<2^{k-1}n^2$.


The term $f(n,k)$ arises as an answer to the following [combinatorics question](https://math.stackexchange.com/questions/602166/decreasing-integers-on-the-blackboard).



Asymptotic growth for $\sum_{i=1}^{n-1}(n-i)\binom{k}{i}$
For $n$ and $k$ fixed, the sequence $\left\{(n-i){k \choose i} \right\}\_{i=1}^{n-1}$
 is strongly unimodal (that is, log concave), or at least a pencil and paper computation convinced me that it is. This allows one to estimate extremely well the sum, $f(n,k)$, particularly when $n$ is large. (The idea is that beyond the mode, the tails go to zero faster than geometric, and so errors are relatively easy to estimate; we just have to look at the behaviour of the ratios of consecutive terms, ....)


