I assume the "between" is inclusive.  The transition matrix $P$ is thus
lower triangular with all entries $1/n$ in the $n$'th row, and you want
$(P^t)_{nj}$ for $1 \le j \le n$.  The ordinary generating function with respect to $t$ is $$ g_{nj}(z) = \sum_{t=0}^\infty z^t (P^t)_{nj} = (I - z P)^{-1}_{nj}$$  For $j=n$ it is easy to see that 
$g_{nn}(z) = \dfrac{n}{n - z}$, i.e. $(P^t)_{nn} = 1/n^t$.
It appears that for $j < n$,

$$g_{nj}(z) = \dfrac{(n-1)!\; z}{(j-1)! \prod_{k=j}^n (k - z)}$$