$$U_{1303}(x, y) = \frac{\left(1 - y\right)^{3} \left(- \frac{2 x}{1 - y} - \sqrt{- \frac{4 x}{1 - y} + 1} + 1\right)^{2}}{4 x^{4}}$$

$$T_{1303}(n, m, k) = \frac{2 k {\binom{4 k + 2 n}{n}} {\binom{k + m + n - 1}{m}}}{2 k + n}$$