Let us consider an asymmetric random walk such that $X_0=0$ and $$X_{t+1}=\begin{cases} X_t+1\qquad p\\X_t-1\qquad 1-p\end{cases}\qquad t\in \mathbb N$$ with $p<0.5$. What is the distribution of the time spent in $[0,+\infty)$ before time $t$?
Formally, this random variable can be written as $$T_t=\sum_{\tau=0}^t 1_{[0,\infty)}(X_\tau).$$
Note that $T_t<\infty$ almost surely, since an asymmetric random walk goes $-\infty$ almost surely and is not recurrent.
My effort:
We could define the auxiliry random variable $$T_{n,t}=\sum_{\tau=0}^t 1_{[n,\infty)}(X_\tau),$$ to have the following recurrency $$T_{n,t}\overset{\mathcal L}=pT_{n-1,t-1}+(1-p)T_{n+1,t-1}+1_{n\ge 0}.$$ where $\overset{\mathcal L}=$ denotes the equality in distribution. This shows some similarity with Pascal's triangle, but I don't know how to solve this recurrency.