Pyramid #992
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
$$T_{992}(n, m, k) = \begin{cases}1&\text{if n=0 , m=0 , k=0} ,\ \\\frac{4^{m} k {\binom{k + 2 n - 1}{n}} {\binom{\frac{k}{2} + 2 m + \frac{n}{2} - 1}{m}}}{k + 2 m + n}&\text{if k even} ,\ \\\frac{k {\binom{k + 2 n - 1}{n}} {\binom{\frac{k}{2} + 2 m + \frac{n}{2} - \frac{1}{2}}{m}} {\binom{k + 4 m + n - 1}{\frac{k}{2} + 2 m + \frac{n}{2} - \frac{1}{2}}}}{\left(k + 2 m + n\right) {\binom{k + 2 m + n - 1}{\frac{k}{2} + m + \frac{n}{2} - \frac{1}{2}}}}&\text{if k odd} \end{cases} $$
Data table
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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