Conjectural closed formulas for self-dual DT invariants of the point quiver

Prove that for the point quiver (one vertex, no arrows) with the trivial slope function and the two self-dual structures corresponding to types $\mathsf{B}$, $\mathsf{C}$, and $\mathsf{D}$, the self-dual DT invariants satisfy the closed formulas $\mathrm{DT}^{\mathrm{sd}}_{\mathsf{B}_n} = (-1)^n \binom{-1/4}{n}$, $\mathrm{DT}^{\mathrm{sd}}_{\mathsf{C}_n} = (-1)^n \binom{-1/4}{n}$, and $\mathrm{DT}^{\mathrm{sd}}_{\mathsf{D}_n} = (-1)^n \binom{1/4}{n}$ for all integers $n \ge 0$.

Background

The authors compute self-dual DT invariants for the simplest quiver (a single vertex with no arrows) and, based on explicit computations using their algorithm, conjecture remarkably simple closed-form expressions in terms of binomial coefficients for the three families corresponding to orthogonal and symplectic types.

They further note a potential conceptual explanation for the equality of the $\mathsf{B}$ and $\mathsf{C}$ series via Langlands duality, and indicate plans to prove the conjecture in future work.