Generating function
$$U_{12}(x, y) = \frac{y + 1}{- x y - x + 1}$$
Explicit formula
$$T_{12}(n, m, k) = \frac{k {\binom{k + n}{m}} {\binom{k + n}{n}}}{k + n}$$
| 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 2 | 1 | 0 | 0 | 0 | 0 |
| 1 | 3 | 3 | 1 | 0 | 0 | 0 |
| 1 | 4 | 6 | 4 | 1 | 0 | 0 |
| 1 | 5 | 1 | 1 | 5 | 1 | 0 |
| 1 | 6 | 15 | 2 | 15 | 6 | 1 |
| 1 | 7 | 21 | 35 | 35 | 21 | 7 |
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