Pyramid #128
Generating function
$$U_{128}(x, y) = \frac{1}{\sqrt{x^{2} y^{2} - 4 x^{2} y - 2 x y + 1}}$$
Explicit formula
$$T_{128}(n, m, k) = \begin{cases}\frac{k {\binom{k + n}{k - 2 m + 2 n}} {\binom{k - m + n}{k}} {\binom{2 k - 2 m + 2 n}{k - m + n}}}{\left(k + n\right) {\binom{2 k}{k}}}&\text{if k odd} ,\ \\\frac{4^{- m + n} k {\binom{k + n}{k - 2 m + 2 n}} {\binom{k - m + n}{k}}}{k + n}&\text{if k even} \end{cases} $$
Data table
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 2 1 0 0 0 0
0 0 6 1 0 0 0
0 0 6 12 1 0 0
0 0 0 3 2 1 0
0 0 0 2 9 3 1
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