Pyramid #448
Generating function
$$U_{448}(x, y) = x + \frac{\sqrt{\frac{4 x}{\left(1 - y\right)^{3}} + \frac{1}{\left(1 - y\right)^{6}}}}{2} + \frac{1}{2 \left(1 - y\right)^{3}}$$
Explicit formula
$$T_{448}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{2 \left(-1\right)^{n - 1} k {\binom{- 2 k + 2 n - 1}{n - 1}}}{n}&\text{if n>0 and m=0} ,\ \\\frac{3 \left(-1\right)^{m - 1} k {\binom{- 3 k + 3 n - 1}{m - 1}} {\binom{2 k - n - 1}{n}}}{m} \end{cases} $$
Data table
1 3 6 1 15 21 28
2 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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