Generating function
$$U_{513}(x, y) = \frac{x + \left(y + 1\right)^{3}}{\left(y + 1\right)^{2}}$$
Explicit formula
$$T_{513}(n, m, k) = {\binom{k}{n}} {\binom{k - 3 n}{m}}$$
| 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | -2 | 3 | -4 | 5 | -6 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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