$$U_{1213}(x, y) = \frac{\frac{x}{\left(1 - y\right)^{3}} + 1}{\left(1 - y\right)^{3}}$$

$$T_{1213}(n, m, k) = {\binom{k}{n}} {\binom{3 k + m + 3 n - 1}{m}}$$