  Recent results by Andrews and Merca on the number of even parts in all
partitions of n into distinct parts, a(n), were derived via generating
functions. This paper extends these results to the number of parts divisible by
k in all the partitions of n for which the multiplicity of each part is
strictly less than k, ak(n). Moreover, a combinatorial proof is provided using
an extension of Glaisher's bijection. Finally, we give the generating functions
for this new family of integer sequences and use it to verify generalized
pentagonal, triangular, and square power recurrence relations.
