The explicit formula is: $P[N_m=n]=(m/n)P[S_n=m]$, where $P[N_m=n]$ is the
probability the position $m$ is hit after exactly $n$ steps, 
$S_n = X_1+X_2+\dots X_n$ and $P[S_n=m]$ is the probablity after $n$ steps
the path to be at the position $m$. This last is well-known and is given
by $P[S_n=m]=(n!/([(n+m)/2]![(n-m)/2]!))p^{(n+m)/2}q^{(n-m)/2}$.
This is true when $n$ and $m$ have the same parity, else the probability
is zero. Additionally $m$ is positive and $n$ is greater than or equal to $m$.
A similar expression can be found for negative $m$.