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  "body": "\n## Statement\n\nLet $n$ be a natural number, and let $C_m$ be the [[Catalan numbers|092DRBFKMG]].\nThen\n$$C_{n+1} = \\sum_{i=0}^{n} C_i\\, C_{n-i}.$$\n\n## Remarks\n\nWith the definition of the node [[Catalan numbers|092DRBFKMG]], the statement holds by definition. Its content is in the combinatorial families that satisfy it.\nExample: $C_4 = C_0 C_3 + C_1 C_2 + C_2 C_1 + C_3 C_0 = 5 + 2 + 2 + 5 = 14$.\nFor [[Dyck words|0R9JR0465V]], the recurrence splits a word at the first return to level $0$.\n\n## Explanations\n\n### L1 human\n\nA structure of size $n + 1$ often splits into a left part of size $i$ and a right part of size $n - i$.\nAdd over all $i$ the number of ways to build the two parts, and you get the next Catalan number.\n",
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  "msc": [
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  "name": "Catalan recurrence",
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