Pyramid #605
Generating function
$$U_{605}(x, y) = \frac{- \sqrt{4 x + \frac{\left(\sqrt{4 y + 1} + 1\right)^{4}}{16}} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}{\sqrt{4 y + 1} + 1}$$
Explicit formula
$$T_{605}(n, m, k) = \begin{cases}0&\text{if n<k} ,\ \\- \frac{\left(-1\right)^{m - 1} k \left(k - 4 n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k + 2 m + 4 n - 1}{m - 1}}}{m n}&\text{if m>0} ,\ \\- \frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0} \end{cases} $$
Data table
0 0 0 0 0 0 0
-1 3 -9 28 -9 297 -1001
1 -7 35 -154 637 -2548 9996
-2 22 -154 88 -4488 21318 -96558
5 -75 675 -475 28875 -15939 82225
-14 266 -2926 24472 -1729 108927 -6317766
42 -966 12558 -121716 98049 -6947472 44799216
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