Generating function
$$U_{264}(x, y) = \frac{\sqrt{1 - 4 x} - \sqrt{- 4 x - 4 y \sqrt{1 - 4 x} + 1}}{2 y \sqrt{1 - 4 x}}$$
Explicit formula
$$T_{264}(n, m, k) = \begin{cases}\frac{k {\binom{k + 2 m - 1}{m}} {\binom{\frac{k}{2} + \frac{m}{2} + n - \frac{1}{2}}{n}} {\binom{k + m + 2 n - 1}{\frac{k}{2} + \frac{m}{2} + n - \frac{1}{2}}}}{\left(k + m\right) {\binom{k + m - 1}{\frac{k}{2} + \frac{m}{2} - \frac{1}{2}}}}&\text{if k odd} ,\ \\\frac{k {\binom{\frac{k}{2} + \frac{m}{2} + n}{n}} {\binom{k + m + 2 n}{\frac{k}{2} + \frac{m}{2} + n}} {\binom{k + 2 m - 1}{m}}}{\left(k + m\right) {\binom{k + m}{\frac{k}{2} + \frac{m}{2}}}}&\text{if k even} \end{cases} $$
1 | 1 | 2 | 5 | 14 | 42 | 132 |
2 | 3 | 12 | 100/3 | 14 | 441 | 1848 |
6 | 1 | 6 | 175 | 98 | 15876/5 | 16632 |
2 | 35 | 28 | 84 | 588 | 19404 | 121968 |
7 | 126 | 126 | 385 | 3234 | 108108 | 792792 |
252 | 462 | 5544 | 1716 | 168168 | 567567 | 4756752 |
924 | 1716 | 24024 | 75075 | 84084 | 2858856 | 26954928 |
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