European Journal of Molecular & Clinical Medicine

Abstract

Lin introduced and studied two new partition functions 𝑷𝑫𝒕(𝒏) and 𝑷𝑫𝑶𝒕(𝒏), which count the number of designated summands in two partition functions 𝑷𝑫(𝒏) and 𝑷𝑫𝑶(𝒏), respectively. In this paper, we prove the generating functions for 𝑷𝑫𝑶𝒕(𝟒𝟖𝒏), 𝑷𝑫𝑶𝒕(𝟒𝟖𝒏+𝟏𝟒), 𝑷𝑫𝑶𝒕(𝟒𝟖𝒏+𝟏𝟔), 𝑷𝑫𝑶𝒕(𝟒𝟖𝒏+𝟐𝟐) and 𝑷𝑫𝑶𝒕(𝟒𝟖𝒏+𝟒𝟔). Also, we find some new congruences and infinite families of congruences modulo 16 and modulo 72 for 𝑷𝑫𝑶𝒕(𝒏).