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  "body": "\n## Statement\n\nLet $n$ be a natural number, and let $C_n$ be the $n$-th [[Catalan number|092DRBFKMG]].\nThen\n$$C_n = \\frac{1}{n + 1} \\binom{2n}{n} = \\frac{(2n)!}{(n + 1)!\\, n!}.$$\n\n## Remarks\n\nThe fraction is an integer, because $C_n = \\binom{2n}{n} - \\binom{2n}{n+1}$.\nThe numerator $\\binom{2n}{n}$ is the [[central binomial coefficient|01WHD9T0V6]].\nExample: $C_4 = \\frac{1}{5} \\binom{8}{4} = \\frac{70}{5} = 14$.\n\n## Explanations\n\n### L1 human\n\nAmong the $\\binom{2n}{n}$ grid paths from $(0, 0)$ to $(n, n)$, exactly one in $n + 1$ stays below the diagonal.\nSo the Catalan number is the central binomial coefficient divided by $n + 1$.\n",
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