A particle lies on the real number line at the origin. For each step taken, the particle moves from its current position a distance (and direction) chosen equi-probably from range $[-1,r]$. However, if the particle would otherwise move left of the origin, the particle is set back at the origin before the next step.

Given $r$, can a closed expression be derived for the probability $P_{n,r}$ of the particle being at the origin after step $n$? Can an expression be derived for the centered variance $E_{n,r}(X^2)$?

This is a variation of a problem featured here, where instead of the particle "dying" upon crossing the origin it is simply set at the origin and allowed to continue. Also, I am asking about the probability of it simply being at the origin at the $n$th step, not whether it has at some point been there.

Being only a graduate, all I've managed to do is conjecture what the expression might be for particular cases. Inspired by the excellent answers to the original problem, I directly simulated the problem to find the probabilities and presumed they were given by a rational expression $\frac{N_r(n)}{(1+r)^n n!}$. Surprisingly, my results indicated that the numerator was likely an integer in these cases. My conjectures as to the numerator for those particular cases are:

• $r=0$: $N_0(n)=n!$ (Known precisely, since $P_{n,0} = 1$ for all $n$)
• $r=1$: $N_1(n) = (2n-1)!!$
• $r\rightarrow\infty$: $N_r(n)\rightarrow (n+1)^{n-1}$

Edit: The conjecture for the case when $r$ gets arbitrarily large has been changed. There was a typo. (I was considering its similarity to Cayley's Formula and typed that out instead!)

At this point I'm lost. Your consideration is appreciated.