Let $$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$ $$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$ $$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\choose k-i}$$ for $k\in\mathbb{N}$, where the binomial coefficients are to be taken as zero if any of the parameters are negative. I am trying to show that $S_k\geq 7/12=S_2$ for all $k$, where $$S_k=\frac{A_k+B_k+C_k}{k{3k-2\choose k}}.$$

The problem is that the formulas are quite complicated, and I can not find a way to work with them. Along the lines of the answers to a previous question, one can show that $S_k\to 3/5$ as $k\to\infty$. Therefore, it would be enough to show that $S_k$ is decreasing from $k=3$, which seems to be true based on the first 10000 values. Ideally, the proof should not involve the computation of $S_k$ for any $k>3$.