The primitive-prime enumeration formula for quantum matrices with three rows

Let $n$ be a positive integer, and let ${\mathcal O}_q(M_{3,n})$ be the quantized coordinate ring of $3\times n$ matrices. Its primitive $\mathcal{H}$-prime ideals are counted by

Primitive-prime enumeration conjecture. The number of primitive $\mathcal{H}$-primes in ${\mathcal O}_q(M_{3,n})$ is

$$\frac{1}{8} \cdot \left( 15\cdot 4^n - 18 \cdot 3^n +13 \cdot 2^n - 6\cdot(-1)^n + 3\cdot (-2)^n\right).$$

This extends the known formulas for $P(1,n)$ and $P(2,n)$ and is part of the proposed search for closed formulas for $P(m,n)$, including the diagonal terms. Its resolution is not indicated in the source.