Generating function
$$U_{333}(x, y) = - \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4 x - 2 \sqrt{4 y + 1} - 2}$$
Explicit formula
$$T_{333}(n, m, k) = \begin{cases}{\binom{k + n - 1}{n}}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{m - 1} \left(k - n\right) {\binom{k + n - 1}{n}} {\binom{- k + 2 m + n - 1}{m - 1}}}{m} \end{cases} $$
| 1 | 1 | -1 | 2 | -5 | 14 | -42 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | -1 | 2 | -5 | 14 | -42 | 132 |
| 1 | -2 | 5 | -14 | 42 | -132 | 429 |
| 1 | -3 | 9 | -28 | 9 | -297 | 1001 |
| 1 | -4 | 14 | -48 | 165 | -572 | 2002 |
| 1 | -5 | 2 | -75 | 275 | -1001 | 364 |
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