Generating function
$$U_{177}(x, y) = \frac{x}{2} + \frac{\left(x + 1\right) \sqrt{y \left(4 x + 4\right) + 1}}{2} + \frac{1}{2}$$
Explicit formula
$$T_{177}(n, m, k) = \begin{cases}{\binom{k}{n}}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{k + m}{n}} {\binom{- k + 2 m - 1}{m - 1}}}{m} \end{cases} $$
| 1 | 1 | -1 | 2 | -5 | 14 | -42 |
| 1 | 2 | -3 | 8 | -25 | 84 | -294 |
| 0 | 1 | -3 | 12 | -5 | 21 | -882 |
| 0 | 0 | -1 | 8 | -5 | 28 | -147 |
| 0 | 0 | 0 | 2 | -25 | 21 | -147 |
| 0 | 0 | 0 | 0 | -5 | 84 | -882 |
| 0 | 0 | 0 | 0 | 0 | 14 | -294 |
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