$$U_{403}(x, y) = \frac{- 2 y - \sqrt{1 - 4 y} + 1}{x \left(2 y + \sqrt{1 - 4 y} - 1\right) + 2 y^{2}}$$

$$T_{403}(n, m, k) = \begin{cases}\frac{k {\binom{k + n}{n}} {\binom{2 k + 2 m + 2 n}{m}}}{k + m + n},\ \\0&\text{if n = 1} ,\ \\0&\text{if n < k } \end{cases} $$