This note analyzes a linear recursion that arises in the computation of the subpacketization level for the multi-message PIR scheme of Banawan and Ulukus. We derive an explicit representation for the normalized subpacketization level $L$, whose smallest integer multiple yields the subpacketization level of the scheme, in terms of the number of servers $N$, the total number of messages $K$, and the number of demand messages $D$. The resulting formula shows that $L$ is a polynomial in $N$ with nonnegative coefficients, and its leading term is $N^{K-D+1}/D$.