Chapter 8.4, Problem 32E

If G(x) is the generating function for the sequence $\left\{a_k\right\}$, what is the generating function for each of these sequences?

a) $2a_0,2a_1,2a_2,2a_3\mathrm{.....}$

b) $0,a_0,a_1,a_2,a_3\mathrm{.....}$ (assuming that terms follow the pattern of all but the first term)

c) $0,0,0,0,a_2,a_3\mathrm{.....}$ (assuming that terms follow the pattern of all but the first four terms)

d) $a_2,a_3,a_{4,}\mathrm{.....}$e) $a_1,2a_2,3a_3,4a_{4,}\mathrm{.....}$ (Hint: Calculus required here.)

f) $a_0^2,2a_0{a}_1,a_1^2+2a_0{a}_2,2a_0{a}_3+2a_1,2a_0{a}_4+2a_1{a}_2+a_2^2,\mathrm{....}$