Pyramid #449
Generating function
$$U_{449}(x, y) = x - \frac{\sqrt{\frac{4 x}{\left(1 - y\right)^{3}} + \frac{1}{\left(1 - y\right)^{6}}}}{2} + \frac{1}{2 \left(1 - y\right)^{3}}$$
Explicit formula
$$T_{449}(n, m, k) = \begin{cases}0&\text{if n<2*k} ,\ \\- \frac{2 \left(-1\right)^{n - 1} k {\binom{- 2 k + 2 n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\- \frac{3 \left(-1\right)^{m - 1} k {\binom{- 3 k + 3 n - 1}{m - 1}} {\binom{2 k - n - 1}{n}}}{m} \end{cases} $$
Data table
0 0 0 0 0 0 0
0 0 0 0 0 0 0
1 -3 3 -1 0 0 0
-2 12 -3 4 -3 12 -2
5 -45 18 -42 63 -63 42
-14 168 -924 308 -693 11088 -12936
42 -63 441 -1911 5733 -126126 21021
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