I have a random walker that starts at $x = 0$. They can step either left or right one unit. However, if they reach $x = -1$ they fall off a cliff (say), so those paths are invalid -- I've heard this called an absorbing barrier. How do I count the number of possible valid paths if the walker has taken $n$ steps?
As a bonus question, how can I count the number of valid paths which end at $x =0$?
$$ \begin{array}{|r|c|c|} \hline &\text{Without absorbing barrier}&\text{With absorbing barrier at $-1$} \\\hline \text{Number of paths of length $n$}&2^n&\displaystyle\binom{n}{\lfloor n/2\rfloor} \\\hline \text{Number of paths of length $n$ ending at $0$*}&\displaystyle\binom{n}{n/2}&\displaystyle\frac1{n/2+1}\binom{n}{n/2} \\\hline \end{array}$$ ${}^*$Assuming $n$ is even.
I present the data this way to highlight the strange coincidence that the lower left entry and the upper right entry are the same (when $n$ is even).