Formulae for the number of partitions of n into at most m parts (using   the quasi-polynomial ansatz)
  The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for pm(n) for any desired m. We do this to demonstrate the power of "rigorous guessing" as facilitated by the quasi-polynomial ansatz. 
Andrew V. Sills and Doron Zeilberger
math.CO math.NT