$$U_{503}(x, y) = \frac{y^{2} \left(2 - 2 \sqrt{- \frac{x \left(1 - \sqrt{1 - 4 y}\right)^{3}}{2 y^{3}} + 1}\right)}{x \left(1 - \sqrt{1 - 4 y}\right)^{2}}$$

$$T_{503}(n, m, k) = \frac{k \left(k + 3 n\right) {\binom{k + 2 n}{n}} {\binom{k + 2 m + 3 n - 1}{m}}}{\left(k + 2 n\right) \left(k + m + 3 n\right)}$$