Pyramid #750
Generating function
$$U_{750}(x, y) = \frac{\left(1 - \frac{4 \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{- x}}{2} \right)}}{3} \right)}}{3}\right)^{2} \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{2 y^{2}}$$
Explicit formula
$$T_{750}(n, m, k) = \begin{cases}\frac{2 \left(-1\right)^{n - 1} k^{2} {\binom{2 k + 2 m}{m}} {\binom{- 2 k + 3 n - 1}{n - 1}}}{n \left(k + m\right)}&\text{if n>0} ,\ \\\frac{k {\binom{2 k + 2 m}{m}}}{k + m}&\text{if n=0} \end{cases} $$
Data table
1 2 5 14 42 132 429
2 4 1 28 84 264 858
-3 -6 -15 -42 -126 -396 -1287
1 2 5 14 42 132 429
-42 -84 -21 -588 -1764 -5544 -18018
198 396 99 2772 8316 26136 84942
-1001 -2002 -5005 -14014 -42042 -132132 -429429
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