I have a function $f(k)$ defined on the set of natural numbers and I managed to show that $f(k)>n-\binom n k(1-n^{-2/3})^{k(k-1)/2}$ for all integers $n\ge k$. I am hoping to get a further estimation that $f(k)>(\frac{k}{3\log k})^{3/2}$ for large $k$.
I am not sure how to do this, but with a graph plotter, I set $n=k^{3/2}$ and from the asymptotic behavior of $(k^{3/2}-\binom {k^{3/2}} k(1-k^{-1})^{k(k-1)/2})/(\frac{k}{3\log k})^{3/2}$ in the graph it seemed to work.
But I couldn't work out how to prove it.
No, it's not true. With $n \approx k^{3/2}$, $n - {n \choose k} (1 - n^{-2/3})^{k(k-1)/2} < 0$ for large $k$ (in fact for any $k \ge 4$).