Pyramid #305
Generating function
$$U_{305}(x, y) = \frac{\sqrt{3} \left(1 - \sqrt{1 - 4 x}\right) \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 x \sqrt{y}}$$
Explicit formula
$$T_{305}(n, m, k) = \frac{k^{2} {\binom{k + 3 m - 1}{m}} {\binom{k + 2 n - 1}{n}}}{\left(k + 2 m\right) \left(k + n\right)}$$
Data table
1 1 3 12 55 273 1428
1 1 3 12 55 273 1428
2 2 6 24 11 546 2856
5 5 15 6 275 1365 714
14 14 42 168 77 3822 19992
42 42 126 504 231 11466 59976
132 132 396 1584 726 36036 188496
Export
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