Problem 1167 (difficulty: 6/10)

Let \(\displaystyle c_0=1\) and \(\displaystyle c_{n+1}=\sum\limits_{k=0}^nc_kc_{n-k}\). (Catalan numbers.) Define \(\displaystyle G(x)=\sum\limits_{n=0}^\infty c_nx^n\) the so called generating function of the Catalan-numbers.

(a) Prove that \(\displaystyle G\) converges in a neigborhood of \(\displaystyle 0\).

(b) Prove that in the (non-empty) interior of the convergence interval \(\displaystyle G(x)=xG^2(x)+1\).

(c) Using b) determine \(\displaystyle G\) and \(\displaystyle c_n\) explicitely.

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