$$U_{514}(x, y) = \frac{\sqrt{4 x + \left(y + 1\right)^{4}} + \left(y + 1\right)^{2}}{2 y + 2}$$

$$T_{514}(n, m, k) = \begin{cases}{\binom{k}{m}}&\text{if n==0} ,\ \\\frac{k {\binom{k - 4 n}{m}} {\binom{k - n - 1}{n - 1}}}{n}&\text{if n>0} \end{cases} $$