$$U_{1267}(x, y) = \frac{1 - \sqrt{- \frac{x \left(1 - \sqrt{1 - 4 y}\right)^{3}}{2 y^{3}} + 1}}{2 x}$$

$$T_{1267}(n, m, k) = \frac{3 k {\binom{k + 2 n - 1}{n}} {\binom{3 k + 2 m + 3 n - 1}{m}}}{3 k + m + 3 n}$$