Truncated sums for the partition function and a problem of Merca (Q2244647)

\textit{G. E. Andrews} and the reviewer derived in [J. Comb. Theory, Ser. A 119, No. 8, 1639--1643 (2012; Zbl 1246.05014)] a truncated version of Euler's pentagonal numbers theorem. As corollary, they obtained the following partition-theoretic interpretation \[ (-1)^{k-1} \sum_{j=1-k}^k (-1)^j\, p\big(n-j(3j-1)/2\big) = M_k(n), \] where \(p(n)\) is the number of partitions of \(n\) and \(M_k(n)\) is the number of partitions of \(n\) in which \(k\) is the least integer that is not a part and there are more parts \(> k\) than there are \(< k\). In this paper, the authors provide a partition-theoretic interpretation of the sum \[ (-1)^{k} \sum_{j=-k}^k (-1)^j\, p\big(n-j(3j-1)/2\big) \] recently requested by the reviewer.