Generating function
$$U_{257}(x, y) = \frac{- \sqrt{1 - 4 x} + \sqrt{2} \sqrt{8 x^{2} y - 2 x - \sqrt{1 - 4 x} + 1} + 1}{4 x}$$
Explicit formula
$$T_{257}(n, m, k) = \begin{cases}1&\text{if m=0 and n = 0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{- k + 2 m - 1}{m - 1}}}{m}&\text{if n=0} ,\ \\\frac{\left(-1\right)^{m + n - 1} k {\binom{- k + 2 m}{m}} {\binom{- k + 2 m - n - 1}{n - 1}}}{n}&\text{if n>0} \end{cases} $$
| 1 | 1 | -1 | 2 | -5 | 14 | -42 |
| 1 | -1 | 3 | -1 | 35 | -126 | 462 |
| 2 | -1 | 0 | 1 | -7 | 378 | -1848 |
| 5 | -2 | 1 | 0 | 35 | -42 | 3234 |
| 14 | -5 | 3 | 0 | 0 | 126 | -231 |
| 42 | -14 | 9 | -2 | 0 | 0 | 462 |
| 132 | -42 | 28 | -1 | 0 | 0 | 0 |
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