$$U_{499}(x, y) = - \frac{1}{\left(1 - y\right)^{2} \left(\frac{x}{\left(1 - y\right)^{4}} - 1\right)}$$

$$T_{499}(n, m, k) = \frac{k {\binom{k + n}{n}} {\binom{2 k + m + 4 n - 1}{m}}}{k + n}$$