Can the expression $\sum_{i=0}^{r} {r\choose i}{n-i+r-1\choose r-1} $ be simplified?

Question

The problem originally was:

Let $r$ be a positive integer, and $p_n$ the number of solutions to the equation: $|x_1|+|x_2|+...+|x_r|=n$ when $x_k$ may be positive, negative or zero.

Find the generating function of $p_n$. The function should be written explicitly (for example, not as a multiplication of two infinite series).

What I did is:

$$(1+2x+2x^2+...)^r=(\frac{1}{1-x}+\frac{x}{1-x})^r=(\frac{1+x}{1-x})^r=\sum_{i=0}^{\infty} \sum_{j=0}^{\infty}{r\choose i}{j+r-1\choose r-1}x^{i+j} $$

Now, for $i+j=n$, we get that the coefficient of $x^n$ is: $$\sum_{i=0}^{\infty} {r\choose i}{n-i+r-1\choose r-1} = \sum_{i=0}^{r} {r\choose i}{n-i+r-1\choose r-1} $$ Something that I am not able to simplify, and I am afraid that my answer (if correct) does not meet the criteria posed in the question.

I have tried to mess around with the expression but couldn't really simplify it.

Any insight will be appreciated.

Answer 1

The generating function which you have found: $$ \left(\frac{1+x}{1-x}\right)^r $$ completely satisfy the restriction:

The function should be written explicitly (for example, not as a multiplication of two infinite series).

What concerns your main question I can suggest another representation of the sum: $$ \sum_{i=0}^r\binom ri\binom{n-i+r-1}{r-1}=\sum_{i=1}^r\binom ri\binom{n-1}{i-1}2^i. $$

I am not sure that this is a real simplification but it looks a little bit nicer.