Generating function
$$U_{784}(x, y) = \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6} \left(- x + \frac{- 2 y - \sqrt{1 - 4 y} + 1}{2 y^{2}}\right)^{2}}$$
Explicit formula
$$T_{784}(n, m, k) = \begin{cases}{\binom{2 k + n - 1}{n}}&\text{if m=0} ,\ \\\frac{2 \left(k - n\right) {\binom{2 k + n - 1}{n}} {\binom{2 k + 2 m - 2 n - 1}{m - 1}}}{m} \end{cases} $$
1 | 2 | 5 | 14 | 42 | 132 | 429 |
2 | 0 | 0 | 0 | 0 | 0 | 0 |
3 | -6 | -3 | -6 | -15 | -42 | -126 |
4 | -16 | 8 | 0 | -4 | -16 | -56 |
5 | -3 | 45 | -1 | 0 | 0 | -5 |
6 | -48 | 12 | -96 | 12 | 0 | 0 |
7 | -7 | 245 | -35 | 175 | -14 | 0 |
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