Let $X_1,X_2,\dots$ be independent identically distributed random variables representing the successive jumps of the particle. Let $M_n$ be the position of the particle at time $n\in\{0,1,\dots\}$, so that $M_0=0$ and $M_n=\max(0,M_{n-1}+X_n)$ for $n\in\{1,2,\dots\}$. Let now $F_n(x):=\mathbb{P}(M_n\le x)$ for real $x$. Then $F_0(x)=\mathrm{I}\{x\ge0\}$ and $F_n(x)=\frac{\mathrm{I}\{x\ge0\}}{1+r}\,\int_{-r}^1 F_{n-1}(x-y)\,dy$ for all real $x$ and $n\in\{1,2,\dots\}$. In particular, the probability in question is $P_{n,r}=\mathbb{P}(M_n=0)=F_n(0)$. 

The work in Mathematica notebook at [n=1,2,3][1] shows that the expression 
$$\frac{\left(\frac{3r}{1+r}\right)_{n-1}}{(1+r)\left(2r\right)_{n-1}},\quad r\geq 1,$$
which you most recently proposed for $P_{n,r}$, is incorrect already for $n=3$ (and $r\ne1$), even though it seems pretty close to the correct one.  

I doubt that there is such a simple closed form expression for $P_{n,r}$. However, using Spitzer's identity and work done in [c.f. of X_+][2], one can obtain an integral expression for the generating function of $P_{n,r}$: 
$$
(*)\qquad \sum_{n=0}^\infty P_{n,r}z^n=
\frac1{\sqrt{1-z}}\exp \Big(\frac1{2 \pi  i}\,\int_0^{\infty } 
\ln\frac{(r+1) u-z (e^{i r u}-e^{-i u})}{(r+1) u-z (e^{i u}-e^{-i r u})} \, \frac{du}{u}\Big)
$$ 
for complex $z$ with $|z|<1$. 
To obtain (*), one can start with formula (23) in [c.f. of X_+][2], analytically extend it to complex $s$ with $\tau:=\Im s\ge0$, and then let $\tau\to\infty$. Alternatively, one can start with the second displayed formula on page 159  (with $\lambda=0$) in 
[Spitzer 1960][3] and then reason as in [c.f. of X_+][4]; here one should also have in mind statement (2.4) on page 155 in [Spitzer 1960][3]. 

In particular, it easily follows from (*) that your expression for $P_{n,r}$ is correct in the symmetric case, when $r=1$, but apparently only in this case. 


  [1]: https://www.dropbox.com/s/w6w2obndgxcdr0y/walk%3Bn%3D1%2C2%2C3.nb?dl=0
  [2]: http://arxiv.org/abs/1309.5928v3
  [3]: http://www.ams.org/journals/tran/1960-094-01/S0002-9947-1960-0111066-X/
  [4]: http://arxiv.org/abs/1309.5928v3