Generating function
$$U_{414}(x, y) = \frac{\sqrt{4 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}}{2} + \frac{\sqrt{4 y + 1}}{4} + \frac{1}{4}$$
Explicit formula
$$T_{414}(n, m, k) = \begin{cases}1&\text{if m=0 and n=0} ,\ \\\frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{k - n - 1}{n}} {\binom{- k + 2 m + 2 n - 1}{m - 1}}}{m}&\text{if m>0} \end{cases} $$
| 1 | 1 | -1 | 2 | -5 | 14 | -42 |
| 1 | -1 | 2 | -5 | 14 | -42 | 132 |
| -1 | 3 | -9 | 28 | -9 | 297 | -1001 |
| 2 | -1 | 4 | -15 | 55 | -2002 | 728 |
| -5 | 35 | -175 | 77 | -3185 | 1274 | -4998 |
| 14 | -126 | 756 | -3822 | 1764 | -77112 | 325584 |
| -42 | 462 | -3234 | 1848 | -94248 | 447678 | -2027718 |
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