Generating function
$$U_{992}(x, y) = \frac{1 - \sqrt{- \sqrt{2} x \sqrt{\frac{1 - \sqrt{1 - 16 y}}{y}} + 1}}{2 x}$$
Explicit formula
$$T_{992}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{4^{m} k {\binom{k + 2 n - 1}{n}} {\binom{\frac{k}{2} + 2 m + \frac{n}{2} - 1}{m}}}{k + 2 m + n}&\text{if k even} ,\ \\\frac{k {\binom{k + 2 n - 1}{n}} {\binom{\frac{k}{2} + 2 m + \frac{n}{2} - \frac{1}{2}}{m}} {\binom{k + 4 m + n - 1}{\frac{k}{2} + 2 m + \frac{n}{2} - \frac{1}{2}}}}{\left(k + 2 m + n\right) {\binom{k + 2 m + n - 1}{\frac{k}{2} + m + \frac{n}{2} - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
1 | 2 | 14 | 132 | 143 | 16796 | 208012 |
1 | 10/3 | 126/5 | 1716/7 | 24310/9 | 352716/11 | 5200300/13 |
2 | 12 | 108 | 1144 | 1326 | 162792 | 208012 |
5 | 35 | 33 | 3575 | 4199 | 52003 | 66861 |
14 | 14 | 154 | 182 | 2261 | 2912168 | 385434 |
42 | 2268/5 | 5148 | 6188 | 8546580/11 | 131047560/13 | 134131032 |
132 | 1848 | 24024 | 31416 | 4178328 | 5653032 | 77654808 |
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