how to determine the sequence generated by these generating functions?

Question

I'm having hard time figuring out the sequence generated by each of these generating functions.

1. $f(x) = {x^4\over (1-x)}$

2. $f(x) = {1\over (3-x)}$

3. $f(x) = {1\over (1-x)+3x^7-11}$

4. $f(x) = {1\over (1+3x)}$

for #1, since $1\over (1-x)$ is $1 + x + x^2 + x^3+ X^4...$ and so on, I have $x^4+x^8+x^12+x^16$ and so on.. but how do I get the sequence(like 0,0,1,1 and so on)?

Any help would be very much appreciated thankyou

Answer 1

As you observe, #1 is just \begin{equation*} \frac{x^4}{1-x} = \sum_{i=0}^\infty x^{4+i} = \sum_{i=0}^\infty a_ix^i, \end{equation*} where $a_i = 1$ if $i\ge 4$, and $a_i = 0$ otherwise. For #2, rewrite $\frac{1}{3-x}$ as \begin{equation*} \frac{1}{3-x} = \frac{1/3}{1-(x/3)} = \frac{1}{3}\cdot\frac{1}{1-(x/3)}. \end{equation*} Then use the power series expansion of $\frac{1}{1-x}$ with $\frac{x}{3}$ in place of $x$.

Answer 2

For 1, you are correct that $\frac 1{1-x}=1+x+x^2+x^3+\dots $. You should multiply it by $x^4$, so there are terms in $x^5$ and $x^6$, etc. rogerl has dealt with 2. For 3 you just update the affected terms in $\frac 1{1-x}$ For 4, define $y=-3x$, write the generating function in $y$ and substitute back in for $x$