We know for Catalan Numbers,there are: $$C_n = \frac{1}{n+1}\binom{2n}{n}$$and $$C_0 = 1,\qquad C_{n+1}=\sum_{i=0}^{n} C_i\, C_{n-i}\,.$$But how can I plug the upper one into the lower one and check if it really works or not through algebraic manipulation?
Catalan Number Proof
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combinatorics
catalan-numbers
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1https://en.wikipedia.org/wiki/Catalan_number#Fifth_proof – 2017-01-20
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0thanks, but they invited Dyck Words to prove it and that one is a inverse proof to my question. Can you come up with a pure algebraic manipulation solution? That would be what I (the professor) am seeking.Thank you in advance. – 2017-01-20