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  "body": "\n## Definition\n\nThe Catalan numbers $C_n$, for natural numbers $n$, are defined by the recursion\n$$C_0 = 1, \\qquad C_{n+1} = \\sum_{i=0}^{n} C_i\\, C_{n-i}.$$\n\n## Remarks\n\nThe first values, for $n = 0, 1, \\dots, 7$, are $1, 1, 2, 5, 14, 42, 132, 429$.\nThe [[closed formula|0NNDQVJ54Q]] is $C_n = \\frac{1}{n+1} \\binom{2n}{n}$.\n$C_n$ counts [[Dyck words|0R9JR0465V]] of semilength $n$ and [[binary trees|02A8EMC83C]] with $n$ nodes, and many other families.\n\n## Explanations\n\n### L1 human\n\nCatalan numbers count many structures that split into a left part and a right part.\nFor example, there are $C_3 = 5$ ways to put brackets on a product $abcd$ of four factors: $((ab)c)d$, $(a(bc))d$, $(ab)(cd)$, $a((bc)d)$ and $a(b(cd))$.\n",
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