Some new results on the number of tagged parts over the partitions with designated summands in which all parts are odd

Let PDOt(n) count the number of tagged parts over the partition function PDO(n), the number of partitions of n with designated summands in which all parts are odd. In this paper, we obtain some new generating functions concerning PDOt(n). Also, we establish several new congruences and infinite families of congruences modulo 16, 48, 144, 288, 1152 and 6912 satisfied by PDOt(n).