Pyramid #916
Generating function
$$U_{916}(x, y) = \frac{x}{3} + \frac{x^{2} + 2 x + 1}{9 \sqrt[3]{\frac{x^{3}}{27} + \frac{x^{2}}{9} + \frac{x \left(27 y + 3\right)}{27} + y + \frac{\sqrt{3} \sqrt{y \left(2 x^{2} + 4 x + 27 y + 2\right)} \left(x + 1\right)}{9} + \frac{1}{27}}} + \sqrt[3]{\frac{x^{3}}{27} + \frac{x^{2}}{9} + \frac{x \left(27 y + 3\right)}{27} + y + \frac{\sqrt{3} \sqrt{y \left(2 x^{2} + 4 x + 27 y + 2\right)} \left(x + 1\right)}{9} + \frac{1}{27}} + \frac{1}{3}$$
Explicit formula
$$T_{916}(n, m, k) = \begin{cases}{\binom{k}{n}}&\text{if m=0 } ,\ \\\frac{2^{m} k {\binom{k - 2 m}{n}} {\binom{k - 2 m - 1}{m - 1}}}{m} \end{cases} $$
Data table
1 2 -8 56 -48 4576 -46592
1 -2 24 -28 336 -41184 512512
0 2 -48 84 -1344 20592 -3075072
0 -2 8 -196 4032 -75504 13325312
0 2 -12 392 -1008 226512 -46638592
0 -2 168 -7056 22176 -5889312 139915776
0 2 -224 1176 -44352 13741728 -373108736
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