Given a random (undirected and unweighted) graph $G$ on $n$ vertices where each of the edges has equal and independent probability $p$ of existing (see Erdős–Rényi model). Fix some vertex $u\in G$. I want to know what is the expected number of vertices a distance $k$ away (where distance means the shortest distance). I can only figure out that we expect there to be $(n-1)p$ vertices a distance 1 away since each of the possible $(n-1)$ edges outgoing from $u$ have probability $p$ of existing, but I am having trouble generalizing from this local property. Maybe someone can handle the task or perhaps point me in the right direction via a paper?
If it makes it easier, assume that $G$ is connected (as it almost surely is as $n\rightarrow \infty$, which is the case I am interested about). As a subproblem of almost equal importance, I would in particular like to know what is the expected longest shortest distance from vertex $u$ and how many vertices are away from vertex $u$ at that distance (subproblem with the maximum possible $k$).
Note: (1) The graph does not have a bound on the degree of the vertex (there could be as many as $n-1$ adjacent vertices to $u$).