$$U_{1083}(x, y) = \frac{\left(- 2 x - \sqrt{1 - 4 x} + 1\right)^{2}}{4 x^{4} \left(1 - y\right)^{3}}$$

$$T_{1083}(n, m, k) = \frac{2 k {\binom{4 k + 2 n}{n}} {\binom{3 k + m - 1}{m}}}{2 k + n}$$