Pyramid #791
Generating function
$$U_{791}(x, y) = \frac{3 x \left(- \sqrt{1 - \frac{y \left(8 - 8 \cos{\left(\frac{2 \arcsin{\left(\frac{3 \sqrt{3} \sqrt{x}}{2} \right)}}{3} \right)}\right)}{3 x}} + 1 - \frac{y \left(4 - 4 \cos{\left(\frac{2 \arcsin{\left(\frac{3 \sqrt{3} \sqrt{x}}{2} \right)}}{3} \right)}\right)}{3 x}\right)}{y^{2} \left(4 - 4 \cos{\left(\frac{2 \arcsin{\left(\frac{3 \sqrt{3} \sqrt{x}}{2} \right)}}{3} \right)}\right)}$$
Explicit formula
$$T_{791}(n, m, k) = \frac{k {\binom{2 k + 2 m}{m}} {\binom{2 k + 2 m + 3 n - 1}{n}}}{k + m + n}$$
Data table
1 2 5 14 42 132 429
2 8 3 112 42 1584 6006
7 36 165 728 315 13464 57057
3 176 91 448 2142 10032 462462
143 91 51 27132 13965 701316 345345
728 4896 2907 163856 892584 473616 2459457
3876 27132 168245 99176 56511 3135132 170080911
Export
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