Generating function
$$U_{41}(x, y) = \frac{x}{2} + \frac{\sqrt{x^{2} + 2 x + 4 y + 1}}{2} + \frac{1}{2}$$
Explicit formula
$$T_{41}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{k {\binom{m + n}{n}} {\binom{k - m - 1}{m + n - 1}}}{m + n} \end{cases} $$
| 1 | 1 | -1 | 2 | -5 | 14 | -42 |
| 1 | -1 | 3 | -1 | 35 | -126 | 462 |
| 0 | 1 | -6 | 3 | -14 | 63 | -2772 |
| 0 | -1 | 1 | -7 | 42 | -231 | 12012 |
| 0 | 1 | -15 | 14 | -105 | 693 | -42042 |
| 0 | -1 | 21 | -252 | 231 | -18018 | 126126 |
| 0 | 1 | -28 | 42 | -462 | 42042 | -336336 |
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