Macmahon’s sums-of-divisors and their connection to multiple Eisenstein series

We give explicit expressions for MacMahon’s generalized sums-of-divisors q-series \(A_r\) $A_r$ and \(C_r\) $C_r$ by relating them to (odd) multiple Eisenstein series. Recently, these sums-of-divisors have been studied in the context of quasimodular forms, vertex algebras, \(N=4\) $N=4$ SU(N) Super–Yang–Mills theory, and the study of congruences of partitions. We relate them to a broader mathematical framework and give explicit expressions for both q-series in terms of Eisenstein series and their odd variants.