Generating function
$$U_{335}(x, y) = \frac{\sqrt{2 x \left(\sqrt{4 y + 1} + 1\right) + \frac{\left(\sqrt{4 y + 1} + 1\right)^{4}}{16}} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}{\sqrt{4 y + 1} + 1}$$
Explicit formula
$$T_{335}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{- k + 2 n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{- k + 2 m - 1}{m - 1}}}{m}&\text{if n=0} ,\ \\\frac{\left(-1\right)^{m - 1} k \left(k - 3 n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k + 2 m + 3 n - 1}{m - 1}}}{m n}&\text{if n>0,m>0} \end{cases} $$
1 | 1 | -1 | 2 | -5 | 14 | -42 |
1 | -2 | 5 | -14 | 42 | -132 | 429 |
-1 | 5 | -2 | 75 | -275 | 1001 | -364 |
2 | -16 | 88 | -416 | 182 | -7616 | 31008 |
-5 | 55 | -385 | 22 | -1122 | 53295 | -241395 |
14 | -196 | 1666 | -11172 | 6517 | -347116 | 173558 |
-42 | 714 | -714 | 54978 | -361284 | 213486 | -1169532 |
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