Pyramid #749
Generating function
$$U_{749}(x, y) = \frac{y^{3} \left(- \frac{2 x \left(1 - \sqrt{1 - 4 y}\right)^{2}}{y^{2}} - 4 \sqrt{- \frac{x \left(1 - \sqrt{1 - 4 y}\right)^{2}}{y^{2}} + 1} + 4\right)}{x^{2} \left(1 - \sqrt{1 - 4 y}\right)^{3}}$$
Explicit formula
$$T_{749}(n, m, k) = \frac{k \left(k + 2 n\right) {\binom{2 k + 2 n}{n}} {\binom{k + 2 m + 2 n - 1}{m}}}{\left(k + n\right) \left(k + m + 2 n\right)}$$
Data table
1 1 2 5 14 42 132
2 6 18 56 18 594 2002
5 25 1 375 1375 5005 182
14 98 49 2156 8918 35672 139944
42 378 2268 11466 5292 231336 976752
132 1452 10164 5808 296208 1406988 6372828
429 5577 44616 284427 1589445 8159151 39507468
Export
expand_less