Congruences modulo $7$ and $11$ for generalized cubic partitions

Abstract: Amdeberhan, Sellers, and Singh introduced the function $a_c(n)$ that counts the number of generalized cubic partitions of $n$, which are partitions of $n$ whose even parts may appear in $c\geq 1$ different colors. Recently, Dockery obtained via modular forms the following isolated congruences modulo $7$ and $11$ for $a_c(n)$, namely $a_5(49n+31)\equiv 0\pmod{7}$ and $a_9(121n+36)\equiv 0\pmod{11}$ for $n\geq 0$. We prove in this short note a generalization of these congruences by employing a result of Ahlgren on the coefficients of a certain product of powers of Euler's product.