Pyramid #79
Generating function
$$U_{79}(x, y) = \frac{x^{2}}{2} + x + \frac{\left(x + 1\right) \sqrt{x^{2} + 2 x + 4 y + 1}}{2} + \frac{1}{2}$$
Explicit formula
$$T_{79}(n, m, k) = \begin{cases}1&\text{if n=0 and m=0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{- k + 2 m - 1}{m - 1}}}{m}&\text{if n=0 and m>0} ,\ \\\frac{2 k {\binom{k - m}{m}} {\binom{2 k - 2 m - 1}{n - 1}}}{n}&\text{if n>0} \end{cases} $$
Data table
1 1 -1 2 -5 14 -42
2 0 2 -8 3 -112 42
1 0 -3 2 -105 504 -231
0 0 4 -4 28 -168 924
0 0 -5 7 -63 462 -3003
0 0 6 -112 126 -11088 84084
0 0 -7 168 -231 24024 -21021
Export
expand_less