New relations for the number of partitions with distinct even parts
From the author's summary and the reviewer: ``The number of partitions of \(n\) wherein even parts are distinct and odd parts are unrestricted, often denoted by ped\((n)\), has been the subject of many recent studies. Cf. [\textit{G. E. Andrews}, Advances in combinatorial mathematics. Berlin: Springer, 31--37 (2009; Zbl 1182.05008); \textit{G. E. Andrews} et al., Ramanujan J. 23, No. 1--3, 169--181 (2010; Zbl 1218.05018); \textit{S.-C. Chen}, Discrete Math. 311, No. 12, 940--943 (2011; Zbl 1232.11110); \textit{E. X. W. Xia}, Electron. J. Comb. 21, No. 4, Research Paper P4.8, 10 p. (2014; Zbl 1298.11099)]. In this paper, the author provides an efficient linear recurrence relation for ped\((n)\). Theorem 1.2 asserts that \[ \sum^\infty_{j=-\infty} (-1)^j\,\text{ped}(n-2j^2)= \begin{cases} 1\quad &\text{if}\quad n=k(k+1)/2,\quad k\in\mathbb{N}_0,\\ 0\quad &\text{otherwise}.\end{cases} \] A simple criterion for deciding whether ped\((n)\) is odd or even is given as a corollary of this result. Some connections with partitions into parts not congruent to \(2\mod 4\), overpartitions and partitions into distinct parts are presented in this context.''
scientific article; zbMATH DE number 6697550
scientific article