Pyramid #264
Generating function
$$U_{264}(x, y) = \frac{\sqrt{1 - 4 x} - \sqrt{- 4 x - 4 y \sqrt{1 - 4 x} + 1}}{2 y \sqrt{1 - 4 x}}$$
Explicit formula
$$T_{264}(n, m, k) = \begin{cases}\frac{k {\binom{k + 2 m - 1}{m}} {\binom{\frac{k}{2} + \frac{m}{2} + n - \frac{1}{2}}{n}} {\binom{k + m + 2 n - 1}{\frac{k}{2} + \frac{m}{2} + n - \frac{1}{2}}}}{\left(k + m\right) {\binom{k + m - 1}{\frac{k}{2} + \frac{m}{2} - \frac{1}{2}}}}&\text{if k odd} ,\ \\\frac{k {\binom{\frac{k}{2} + \frac{m}{2} + n}{n}} {\binom{k + m + 2 n}{\frac{k}{2} + \frac{m}{2} + n}} {\binom{k + 2 m - 1}{m}}}{\left(k + m\right) {\binom{k + m}{\frac{k}{2} + \frac{m}{2}}}}&\text{if k even} \end{cases} $$
Data table
1 1 2 5 14 42 132
2 3 12 100/3 14 441 1848
6 1 6 175 98 15876/5 16632
2 35 28 84 588 19404 121968
7 126 126 385 3234 108108 792792
252 462 5544 1716 168168 567567 4756752
924 1716 24024 75075 84084 2858856 26954928
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