On the Number of Partitions of $n$ into Exactly $m$ Parts Whose Even Parts are Distinct

Let $ped(n)$ be the number of partitions of $n$ whose even parts are distinct and whose odd parts are unrestricted. For a positive integer $m$, let $ped(n, m)$ be the number of all possible partitions of the number $n$ into exactly $m$ parts whose even parts are distinct and whose odd parts are unrestricted. In this paper, we give new recurrence formulas for $ped(n,m)$ as well as explicit formulas for $ped(n, m)$, when $m=2, 3$ and $m=4$. For a positive integer $q$ and $j\in\{0,1,2,\ldots,q-1\}$, we also give a recurrence formula for $p_{q,j}(n,m)$ the number of partitions of $n$ into $m$ parts such that the parts congruent to $-j$ modulo $q$ are distinct, where other parts are unrestricted.