Using the generating function \(xC(x)^2-C(x)+1=0\), show that \(C_4=14\) from \(C_0=1\), \(C_1=1\), \(C_2=2\), \(C_3=5\), and check the result with the closed formula.
The equation gives \(C_{n+1}=\sum_{k=0}^{n}C_kC_{n-k}\), and with \(n=3\) one gets 14, matching \(\frac{1}{5}\binom{8}{4}=14\); the option 28 doubles the result by counting the symmetric sum twice.
The Catalan number \(C_n\) counts, among many other structures, the triangulations of a convex polygon with \(n+2\) sides and the Dyck paths of length \(2n\).