Pyramid #49
Generating function
$$U_{49}(x, y) = \frac{x^{2} - 2 x - \sqrt{x^{4} - 4 x^{3} + 6 x^{2} - 4 x + y \left(4 x - 4\right) + 1} + 1}{2 y}$$
Explicit formula
$$T_{49}(n, m, k) = \begin{cases}\frac{k {\binom{k + 2 m - 1}{m}} {\binom{k + 3 m + n - 1}{n}}}{k + m}&\text{if m=0} ,\ \\\frac{k {\binom{k + 2 m - 1}{m}} {\binom{k + 3 m + n - 1}{n}}}{k + m} \end{cases} $$
Data table
1 1 2 5 14 42 132
1 4 14 5 182 672 2508
1 1 56 275 1274 5712 2508
1 2 168 11 637 34272 17556
1 35 42 3575 2548 162792 96558
1 56 924 1001 86632 651168 4441668
1 84 1848 25025 259896 2279088 17766672
Export
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