What is a good approximation to $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{\binom{k(k-1)/2}{i}}$$ $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{(\log k)^2}-1-i)\binom{k(k-1)/2}{i}},\quad\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{\sqrt{k}}-1-i)\binom{k(k-1)/2}{i}}$$ $$\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{{k}}-1-i)\binom{k(k-1)/2}{i}}$$ as a function of $k,i$ when $i\in\{1,2,\dots,k-1,k\}$?
I am looking to see the asymptotics of $$\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{\binom{k(k-1)/2}{i}}$$ $$\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{(\log k)^2}-1-i)\binom{k(k-1)/2}{i}},\quad\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{\sqrt{k}}-1-i)\binom{k(k-1)/2}{i}}$$ $$\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{{k}}-1-i)\binom{k(k-1)/2}{i}}$$
I simulated in mathematica functions $$\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{(\log k)^2}-1-i)\binom{k(k-1)/2}{i}},\quad\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{\sqrt k}-1-i)\binom{k(k-1)/2}{i}}$$ and both seems to initially decrease and then rapidly increases starting around $k=200$ to $300$.
However $$\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{{k}}-1-i)\binom{k(k-1)/2}{i}}$$ seems to keep decreasing.
I suspect $$\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{(\log k)^a}-1-i)\binom{k(k-1)/2}{i}},\quad\displaystyle\sum_{i=1}^k\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{k^{1/a}}-1-i)\binom{k(k-1)/2}{i}}$$ both to initially decrease and then increase if $a>1$ while they keep decreasing at $a=1$. Is there an easy method to show this?
This is not an answer but it is too long for a comment.
Considering $$S(k)=\sum _{i=1}^k \frac{(i-1)! \binom{k}{i}^2}{\binom{\frac{1}{2} k(k-1) }{i}}$$ I have not been able to establish asymptotics but numerical simulations show that $\log\big(S(k)\big)$ varies almost linearly with $k$.
Based on values computed for $k=100,200,\cdots,2000$ an empirical curve fit has been done using $$\log\big(S(k)\big)=a+b\, k +c \,\log(k)+d\, k\,\log(k)$$ for which all parameters are highly significant. $$\{a=2.52691,b= 0.107636,c= -0.534135,d= -0.0000305592\}$$