Generating Function for Integer Compositions The Full Question
(a) What is the generating function for the number of integer compositions with $2$
parts?
(b) What is the generating function for the number of integer compositions with $3$
parts?
(c) What is the generating function for the number of integer compositions with $2$
or $3$ parts?
My Work
Part A
Let $n =$ the number we are finding compositions for
We are trying to solve all the solutions for $x_1 + x_2 = n$ where $ x_1,x_2\geq 1$ and $n\geq2$
The general case for this is $n-1$
Which means our generating function is given by $\sum_{n=2}^{\infty}(k-1)x^{k}$
Part B
Same thought process as part A
Part C
$C = A \cup B$ so $C(x)=A(x)+B(x)$
My Problem
I can't think of a closed form of the generating function I came up with in part A. It doesn't seem to match any of the basic generating functions I was given (These kind of problems usually reduce to one of those). Does anyone know a closed form for that GF?
The Generating Functions I know
 
 A: Note: Since your work is quite ok, we concentrate on your question and add a small remark to part B.)

The generating function
  $\sum_{n=2}^{\infty}(n-1)x^n$
  is a variation of case 8.)
Indeed, we can write:
  \begin{align*}
\sum_{n=2}^{\infty}(n-1)x^n&=\sum_{n=0}^{\infty}(n+1)x^{n+2}\\
&=x^2\sum_{n=0}^{\infty}(n+1)x^{n}\\
&=x^2\sum_{n=0}^{\infty}\binom{n+1}{n}x^{n}\tag{1}\\
&=x^2\sum_{n=0}^{\infty}\binom{-2}{n}(-x)^{n}\tag{2}\\
&=\left(\frac{x}{1-x}\right)^2\tag{3}
\end{align*}

Comment:


*

*In (1) we write $(n+1)$ as binomial coefficient $\binom{n+1}{n}$

*In (2) we use the identity $\binom{-k}{n}=\binom{k+n-1}{k}(-1)^n$

*In (3) we use the generating function according to your table entry 8.)

Part B: 
We observe: Since $$\frac{x}{1-x}=x^1+x^2+x^3+\ldots$$ 
  carries as exponents the possible solutions $1,2,3,\ldots$ of one $x_i$ in
  $$x_1+x_2+x_3+\ldots+x_k=n\qquad k\geq 1, n\geq k$$
  the number of compositions of $n$ of $k$ variables $x_1,x_2,\ldots,x_k$ is the coefficient of $x^n$ of
  $$\left(\frac{x}{1-x}\right)^k$$

In case $k=3$ we obtain therefore the generating function
$$\left(\frac{x}{1-x}\right)^3$$
