  The $K$-type formulas of the space of $K$-finite solutions to the Heisenberg
ultrahyperbolic equation $\square_sf=0$ for the non-linear group
$\widetilde{SL}(3,\mathbb{R})$ are classified. This completes a previous study
of Kable for the linear group $SL(m,\mathbb{R})$ in the case of $m=3$, as well
as generalizes our earlier results on a certain second order differential
operator. As a by-product we also show several properties of certain sequences
$\{P_j(x;y)\}_{j=0}^\infty$ and $\{Q_j(x;y)\}_{j=0}^\infty$ of tridiagonal
determinants, whose generating functions are given by local Heun functions. In
particular, it is shown that these sequences satisfy a certain
arithmetic-combinatorial property, which we refer to as a palindromic property.
We further show that classical sequences of Cayley continuants
$\{\mathrm{Cay}_j(x;y)\}_{j=0}^\infty$ and Krawtchouk polynomials
$\{\mathcal{K}_j(x;y)\}_{j=0}^\infty$ also admit this property. In the end a
new proof of Sylvester's formula for certain tridiagonal determinant
$\mathrm{Sylv}(x;n)$ is provided from a representation theory point of view.
