In this paper we consider the hypersum polynomials P_k^(m)(n) = Σ_r=0^k+m+1 c^r_r,m n^r, and give an explicit formula for the coefficients c^r_{r ,m}. We show that the c^r_{r ,m} satisfy a generalized Akiyama-Tanigawa recurrence relation, thus extending some previous results due to Inaba. We also give a number of identities involving the Stirling numbers of the first and second kinds, as well as Bernoulli and harmonic numbers.