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

Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k