Generating function
$$U_{750}(x, y) = \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- x}}{2} \right)}}{3} \right)}}{3}\right)^{2} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{2 y^{2}}$$
Explicit formula
$$T_{750}(n, m, k) = \begin{cases}\frac{2 \left(-1\right)^{n - 1} k^{2} {\binom{2 k + 2 m}{m}} {\binom{- 2 k + 3 n - 1}{n - 1}}}{n \left(k + m\right)}&\text{if n>0} ,\ \\\frac{k {\binom{2 k + 2 m}{m}}}{k + m}&\text{if n=0} \end{cases} $$
1 | 2 | 5 | 14 | 42 | 132 | 429 |
2 | 4 | 1 | 28 | 84 | 264 | 858 |
-3 | -6 | -15 | -42 | -126 | -396 | -1287 |
1 | 2 | 5 | 14 | 42 | 132 | 429 |
-42 | -84 | -21 | -588 | -1764 | -5544 | -18018 |
198 | 396 | 99 | 2772 | 8316 | 26136 | 84942 |
-1001 | -2002 | -5005 | -14014 | -42042 | -132132 | -429429 |
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