Formula + recurrence
C_n = C(2n, n) - C(2n, n+1). Recurrence: C_{n+1} = sum_{i=0}^n C_i · C_{n-i}.
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Counted objects
Binary trees with n internal nodes. Balanced parentheses. Triangulations of (n+2)-gon. Monotonic lattice paths without crossing diagonal.
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Reflection principle
C_n = (# lattice paths) - (# bad paths). Reflection maps bad paths to unrestricted → bijection proof.