Inequality with choose function: $1\sqrt{\binom n1} + 2\sqrt{\binom n2}+3\sqrt{\binom n3}+\cdots+n\sqrt{\binom nn} < \sqrt{{2^{n-1}}{n^3}}$

Question

From the 1987 Spanish Mathematical Olympiad: Prove, for all natural numbers $n$ with $n > 1$, that $1\sqrt{n\choose1} + 2\sqrt{n\choose2}+3\sqrt{n\choose3}+\cdots+n\sqrt{n\choose{n}} < \sqrt{{2^{n-1}}{n^3}}$.

I tried using some estimates on the size of $n\choose k$, like ${n \choose k} \leq \frac{n^k}{k!}$, but nothing seemed to work, mainly due to all the choose functions being enclosed within square-roots.

Answer 1

By C-S $$1\sqrt{n\choose1} + 2\sqrt{n\choose2}+3\sqrt{n\choose3}+\cdots+n\sqrt{n\choose{n}}\leq\sqrt{(1^2+2^2+...+n^2)(2^n-1)} \leq\sqrt{{2^{n-1}}{n^3}}$$