Pyramid #784
Generating function
$$U_{784}(x, y) = \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{3}}{8 y^{6} \left(- x + \frac{- 2 y - \sqrt{1 - 4 y} + 1}{2 y^{2}}\right)^{2}}$$
Explicit formula
$$T_{784}(n, m, k) = \begin{cases}{\binom{2 k + n - 1}{n}}&\text{if m=0} ,\ \\\frac{2 \left(k - n\right) {\binom{2 k + n - 1}{n}} {\binom{2 k + 2 m - 2 n - 1}{m - 1}}}{m} \end{cases} $$
Data table
1 2 5 14 42 132 429
2 0 0 0 0 0 0
3 -6 -3 -6 -15 -42 -126
4 -16 8 0 -4 -16 -56
5 -3 45 -1 0 0 -5
6 -48 12 -96 12 0 0
7 -7 245 -35 175 -14 0
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