What you are asking for is possible $f$-vectors of [Clique Complexes][1]. The [Kruskal-Katona Theorem][2] characterizes $f$-vectors of simplicial complexes; so, it applies here but is no longer a complete characterization. Your problem is open according to [this paper][3]. The linked paper and references there in contain partial results.

**Edit:** This answer assumes the number of $k$-cliques of $G$ is $a_k$ for $1 \leq k \leq n$ and $0$ for $k > n$. Another interpretation of the question would be that the number of $k$-cliques of $G$ is $a_k$ for $1 \leq k \leq n$ and there is no restriction on the number of $k$-cliques for $k > n$. The OP has expressed interest in both variations in the comments.


  [1]: https://en.m.wikipedia.org/wiki/Clique_complex
  [2]: https://en.m.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem
  [3]: http://arxiv.org/pdf/1403.6210.pdf