Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$ $$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$ $$C_k = \frac{\sum_{i=1}^k(2k-2i-1){2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}},$$ for $k\in\mathbb{N}$. The binomial coefficient is to be taken as zero if any of its parameters are negative.

Based on the values of $S_k:=A_k+B_k+C_k$ for $k=1,\dots,10000$, I conjecture that $S_k$ is decreasing from $k=3$ and $S_k\to2/3$ as $k\to\infty$. I have been struggling with a formal mathematical proof for a few days, and I hope that somebody could point me to the right direction.

Note that based on their first 10000 values, I observed that $A_k,B_k$ and $C_k$ seem to tend to $2/9$ as $k\to\infty$, furthermore, $A_k$ and $B_k$ are decreasing whereas $C_k$ is increasing from $k=3$. Also note that $B_k+C_k$ is simply $$\frac{\sum_{i=1}^k(2k-i-1){2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}},$$

but I found it interesting that all of $A_k,B_k$ and $C_k$ tend to $2/9$. It may be better to handle $B_k+C_k$ as a unite.