Generating function
$$U_{523}(x, y) = - \frac{1}{\left(1 - y\right)^{3} \left(x - \frac{1}{\left(1 - y\right)^{2}}\right)}$$
Explicit formula
$$T_{523}(n, m, k) = {\binom{k + n - 1}{n}} {\binom{k + m - 2 n - 1}{m}}$$
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | -1 | 0 | 0 | 0 | 0 | 0 |
| 1 | -3 | 3 | -1 | 0 | 0 | 0 |
| 1 | -5 | 1 | -1 | 5 | -1 | 0 |
| 1 | -7 | 21 | -35 | 35 | -21 | 7 |
| 1 | -9 | 36 | -84 | 126 | -126 | 84 |
| 1 | -11 | 55 | -165 | 33 | -462 | 462 |
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