$$U_{377}(x, y) = \frac{\sqrt{\sqrt{1 - 4 y} \left(- 16 x y + 4 x\right) + 1} + 1}{2 \sqrt{1 - 4 y}}$$

$$T_{377}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{\left(-1\right)^{m - 1} \cdot 4^{m} k {\binom{- \frac{k}{2} + n - 1}{m - 1}}}{2 m}&\text{if m>0 and n=0} ,\ \\\frac{\left(-1\right)^{n - 1} \cdot 4^{m} k {\binom{- k + 2 n - 1}{n - 1}} {\binom{\frac{k}{2} + m - \frac{3 n}{2} - 1}{m}}}{n}&\text{if n>0} \end{cases} $$