Modularity of moments of reciprocal sums for partitions into distinct parts

In this paper, we determine modularity properties of the generating function of $s_k(n)$ which sums $k$-th power of reciprocals of parts throughout all of the partitions of $n$ into distinct parts. In particular, we show that the generating function for $s_k(n)$ is related to Maass Eisenstein series and sesquiharmonic Maass forms.