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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\...

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(...