Pyramid #655
Generating function
$$U_{655}(x, y) = \frac{y \left(3 - 3 \sqrt{- \frac{32 \sqrt{3} x \sin^{3}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{9 y^{\frac{3}{2}}} + 1}\right)}{8 x \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}$$
Explicit formula
$$T_{655}(n, m, k) = \frac{k \left(k + 3 n\right) {\binom{k + 2 n}{n}} {\binom{k + 3 m + 3 n - 1}{m}}}{\left(k + 2 n\right) \left(k + 2 m + 3 n\right)}$$
Data table
1 1 3 12 55 273 1428
1 4 18 88 455 2448 13566
2 14 84 49 2856 16758 99176
5 5 375 255 16625 10626 67275
14 182 1638 1274 92092 63882 4322682
42 672 7056 61824 4914 3683232 26581632
132 2508 30096 293436 254562 20525472 157582656
Export
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