On recent Partition function of Kaur and Rana

Recently, Kaur and Rana introduced the partition function denoted by $\rho (n)$, where the largest part $\lambda$ appears exactly once, and the remaining parts constitute a partition of $\lambda$. In this paper, we establish new generating functions for certain variants of $\rho (n)$. Further, we obtain a linear recurrence relation for our new generating function.