Pyramid #484
Generating function
$$U_{484}(x, y) = \frac{\frac{2 x}{3} + \frac{\left(y + 1\right)^{4}}{9}}{\sqrt[3]{\frac{x^{2}}{2 \left(y + 1\right)^{2}} + \frac{\sqrt{3} x \sqrt{x \left(27 x + 4 \left(y + 1\right)^{4}\right)}}{18 \left(y + 1\right)^{2}} + \frac{x \left(y + 1\right)^{2}}{3} + \frac{\left(y + 1\right)^{6}}{27}}} + \frac{\left(y + 1\right)^{2}}{3} + \sqrt[3]{\frac{x^{2}}{2 \left(y + 1\right)^{2}} + \frac{\sqrt{3} x \sqrt{x \left(27 x + 4 \left(y + 1\right)^{4}\right)}}{18 \left(y + 1\right)^{2}} + \frac{x \left(y + 1\right)^{2}}{3} + \frac{\left(y + 1\right)^{6}}{27}}$$
Explicit formula
$$T_{484}(n, m, k) = \begin{cases}{\binom{2 k}{m}}&\text{if n=0} ,\ \\\frac{2 k {\binom{2 k - 4 n}{m}} {\binom{2 k - 2 n - 1}{n - 1}}}{n}&\text{if n>0} \end{cases} $$
Data table
1 2 1 0 0 0 0
2 -4 6 -8 1 -12 14
-3 18 -63 168 -378 756 -1386
1 -1 55 -22 715 -2002 5005
-42 588 -441 2352 -9996 359856 -1139544
198 -3564 33858 -22572 118503 -5214132 19987506
-1001 22022 -253253 2026024 -1266265 6584578 -29630601
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