Generating function
$$U_{1444}(x, y) = \frac{\left(y + 1\right)^{3}}{\left(- x \left(y + 1\right)^{2} + 1\right)^{3}}$$
Explicit formula
$$T_{1444}(n, m, k) = {\binom{3 k + 2 n}{m}} {\binom{3 k + n - 1}{n}}$$
| 1 | 3 | 3 | 1 | 0 | 0 | 0 |
| 3 | 15 | 3 | 3 | 15 | 3 | 0 |
| 6 | 42 | 126 | 21 | 21 | 126 | 42 |
| 1 | 9 | 36 | 84 | 126 | 126 | 84 |
| 15 | 165 | 825 | 2475 | 495 | 693 | 693 |
| 21 | 273 | 1638 | 6006 | 15015 | 27027 | 36036 |
| 28 | 42 | 294 | 1274 | 3822 | 84084 | 14014 |
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