Generating function
$$U_{40}(x, y) = \frac{y}{2} + \frac{\sqrt{4 x + y^{2} + 2 y + 1}}{2} + \frac{1}{2}$$
Explicit formula
$$T_{40}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{k {\binom{m + n}{m}} {\binom{k - n - 1}{m + n - 1}}}{m + n} \end{cases} $$
| 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | -1 | 1 | -1 | 1 | -1 | 1 |
| -1 | 3 | -6 | 1 | -15 | 21 | -28 |
| 2 | -1 | 3 | -7 | 14 | -252 | 42 |
| -5 | 35 | -14 | 42 | -105 | 231 | -462 |
| 14 | -126 | 63 | -231 | 693 | -18018 | 42042 |
| -42 | 462 | -2772 | 12012 | -42042 | 126126 | -336336 |
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