Calculations of _integral_ homology of $\mathcal{M}_{g, n}$ occur in [Abhau, Bodigheimer, Ehrenfried][1] (p. 3) or [Godin][2] (p. 4). Of course, in the stable range (degrees $3* \leq 2g$) the rational cohomology is entirely known (by Mumford's conjecture, now theorem), and is a polynomial algebra on a single generator in each even degree. Thus the Betti numbers in this range are given by partition functions.

I should mention that Godin's results were obtained using the complex of fat graphs, which is probably equal to Kontsevich's graph complex for the associative operad.


  [1]: https://www.math.uni-bonn.de/people/cfb/PUBLICATIONS/homology-computation.ps
  [2]: https://arxiv.org/PS_cache/math/pdf/0501/0501304v2.pdf