Generating function
$$U_{93}(x, y) = \frac{- 2 x - 2 y - \sqrt{- 4 x - 4 y + 1} + 1}{2 x + 2 y}$$
Explicit formula
$$T_{93}(n, m, k) = \begin{cases}0&\text{if n=0,m=0} ,\ \\\frac{k {\binom{m + n}{m}} {\binom{2 m + 2 n}{- k + m + n}}}{m + n} \end{cases} $$
0 | 1 | 2 | 5 | 14 | 42 | 132 |
1 | 4 | 15 | 56 | 21 | 792 | 3003 |
2 | 15 | 84 | 42 | 198 | 9009 | 4004 |
5 | 56 | 42 | 264 | 15015 | 8008 | 408408 |
14 | 21 | 198 | 15015 | 1001 | 612612 | 352716 |
42 | 792 | 9009 | 8008 | 612612 | 4232592 | 27159132 |
132 | 3003 | 4004 | 408408 | 352716 | 27159132 | 192203088 |
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