Generating function
$$U_{331}(x, y) = - \frac{x}{2} + \frac{y}{2} - \frac{\sqrt{x^{2} - 2 x + y^{2} + y \left(2 x + 2\right) + 1}}{2} + \frac{1}{2}$$
Explicit formula
$$T_{331}(n, m, k) = \begin{cases}0&\text{if m<k} ,\ \\0&\text{if n<k} ,\ \\- \frac{\left(-1\right)^{n} k {\binom{k - m - 1}{n}} {\binom{k - n - 1}{m - 1}}}{m} \end{cases} $$
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 1 | -1 | 1 | -1 | 1 |
| 0 | -1 | 3 | -6 | 1 | -15 | 21 |
| 0 | -1 | 6 | -2 | 5 | -105 | 196 |
| 0 | -1 | 1 | -5 | 175 | -49 | 1176 |
| 0 | -1 | 15 | -105 | 49 | -1764 | 5292 |
| 0 | -1 | 21 | -196 | 1176 | -5292 | 19404 |
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