Chapter 8.4, Problem 8E

In Exercises 3-8, by a closed form we mean an algebraic expression not involving a summation over a range of values or the use of ellipses.

For each of these generating functions, provide a closed formula for the sequence it determines.

a) ${\left(x^2+1\right)}^3$b) ${\left(3x-1\right)}^3$

c) $1\left(1-2x^2\right)$

d) $x^2{\left(1-x\right)}^3$

e) $x-1+\left(1/\left(1-3x\right)\right)$

f) $\left(1+x^3\right)/{\left(1+x\right)}^3$g) $x/\left(1+x+x^2\right)$

h) $e^{3x2}-1$