a the generating function gz for a sequence gn is

Question

(a) The generating function G(z) for a sequence g_n is given by

G(z) = 1 - 2z/(1 + 3z)3

Give an explicit formula for g_n.

(b) For the sequence gn in the previous part consider the sequence hn defined by hn = 2ng0 + 2n-1g1 + · · ·+gn for all n > 0. Give a closed formula for the generating function H(z) of the sequence hn.

(c) Hence find an explicit formula for h_n.