The goal is to find a closed form for $$S_n:=\sum_{k=0}^n\min(k,n-k)\binom nk,\quad n\geq 0.$$ This question is from Concrete Mathematics.
The problem basically reduces to finding $$R_n=\sum_{k=0}^{n/2}k\binom nk,$$
since $$S_n=2R_n+\mathbb 1_{\text{$n$ is even}}\cdot\binom{n}{n/2}\cdot\frac n2.$$
It is known how to find $R_n$ in case the sum went up to $n$ instead of $n/2$. Any clues on how to proceed here?
