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 $k\in\mathbb{N}$. Binomial coefficients are to be taken as zero if any of the parameters are negative.

I want to prove that $S_k:=A_k+B_k+C_k$ 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 can point me to the right direction.

Note that based on their first 10000 values, the above statements seem to hold, and $A_k,B_k$ 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

$$\frac{\sum_{i=1}^k(2k-i-1){2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}.$$

The reason for not making this simplification is that 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.

Motivation: This question is related to a preceding question. In the setting explained in the other question, $S_k$ is the probability of the marked red ball staying red. The above statements are already proven in an excellent answer to the preceding question. The aim of the present question is to give a new proof using $A_k,B_k,C_k$.