Exact enumeration of lozenge tilings of a triangular region

We prove that the number of lozenge tilings of a certain triangular region ${\mathcal{T}}_n$ is given by the formula $$T_n=\prod _{\begin{array}{c}1\le a<b\le 3n+2\\ (a,b)≠(n+1,2n+2)\end{array}}{\mid 1+{\zeta }^a+{\zeta }^b\mid }^{1\mathrm{/}3},$$ where $\zeta =e^{2\pi i\mathrm{/}(3n+3)}$. This answers a question of Ciucu and Krattenthaler, both by finding the exact formula and by explaining why $T_n$ has many prime factors. The proof reduces the lozenge tiling enumeration problem to evaluating the determinant of the bipartite adjacency matrix $M_n$ of the dual graph of ${\mathcal{T}}_n$, and then evaluates this determinant by diagonalising $M_n$.