Pyramid #335
Generating function
$$U_{335}(x, y) = \frac{\sqrt{2 x \left(\sqrt{4 y + 1} + 1\right) + \frac{\left(\sqrt{4 y + 1} + 1\right)^{4}}{16}} + \frac{\left(\sqrt{4 y + 1} + 1\right)^{2}}{4}}{\sqrt{4 y + 1} + 1}$$
Explicit formula
$$T_{335}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{- k + 2 n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{- k + 2 m - 1}{m - 1}}}{m}&\text{if n=0} ,\ \\\frac{\left(-1\right)^{m - 1} k \left(k - 3 n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k + 2 m + 3 n - 1}{m - 1}}}{m n}&\text{if n>0,m>0} \end{cases} $$
Data table
1 1 -1 2 -5 14 -42
1 -2 5 -14 42 -132 429
-1 5 -2 75 -275 1001 -364
2 -16 88 -416 182 -7616 31008
-5 55 -385 22 -1122 53295 -241395
14 -196 1666 -11172 6517 -347116 173558
-42 714 -714 54978 -361284 213486 -1169532
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