On the number of partitions of n into exactly m parts whose even parts are distinct

Let ped(n) ( 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, ( 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, ( n, m ) as well as explicit formulas for ped(n, ( n, m ), when m = 2,3 , 3 and m = 4. For a positive integer q and j is an element of {0, , 1, , 2, , ... , 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 j modulo q are distinct where other parts are unrestricted.