Catalan Numbers: Formula, Applications & Example

SOLVED:(Calculus required) Let {Cn} be the sequence of Catalan numbers, that is, the solution to the recurrence relation Cn=∑k=0^n-1 Ck Cn-k-1 with C0=C1=1 (see Example 5 in Section 8.1) a) Show that if G(x) is the generating function for the sequence of Catalan numbers, then x G(x)^2-G(x) 1=0 . Conclude (using the initial conditions) that G(x)=(1-√(1-4 x)) /(2 x) b) Use Exercise 42 to conclude that G(x)=∑n=0^∞ (1)/(n 1)( 2 n n) x^n so that Cn=(1)/(n 1)( 2 n n) c) Show that Cn ≥2^n-