How to find the tight bound for this binomial series $\binom{n}{1}*1+\binom{n}{2}*2,+\ldots,+ \binom{n}{n/2}*(n/2)$
I have found $n2^n$ by this rule $\binom{n}{1}+\binom{n}{2},+\ldots,+ \binom{n}{n/2}=2^n/2$. So, by multyplying $n/2$ with $2^n/2$ we get $O(n2^n)$. But, I feel that my bound is not tight. Please let me know if other solutions are possible.
Using the identity $\binom{n}{k}=\frac{n}{k}\binom{n-1}{k-1}$, we get (assuming $n$ is even) $$ \sum_{k=1}^{\frac{n}{2}}k\binom{n}{k}=\sum_{k=1}^{\frac{n}{2}}k\frac{n}{k}\binom{n-1}{k-1}=n\sum_{j=0}^{\frac{n}{2}-1}\binom{n-1}{j}. $$ Can you go from here? (A similar case works for $n$ odd)