$$U_{1234}(x, y) = \frac{\left(\frac{x}{\left(1 - y\right)^{3}} + 1\right)^{3}}{\left(1 - y\right)^{3}}$$

$$T_{1234}(n, m, k) = {\binom{3 k}{n}} {\binom{3 k + m + 3 n - 1}{m}}$$