Generating function
$$U_{1293}(x, y) = \frac{\left(1 - \sqrt{1 - 4 y}\right)^{3} \sqrt{\frac{2 x \left(1 - \sqrt{1 - 4 y}\right)}{y} + 1}}{8 y^{3}}$$
Explicit formula
$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{1293}(n, m, k) = \frac{\left(3 k + n\right) \operatorname{Tsqrt}{\left(n,k \right)} {\binom{3 k + 2 m + n - 1}{m}}}{3 k + m + n}$$
1 | 3 | 9 | 28 | 9 | 297 | 1001 |
2 | 8 | 28 | 96 | 33 | 1144 | 4004 |
-2 | -10 | -40 | -150 | -550 | -2002 | -7280 |
4 | 24 | 108 | 440 | 1716 | 6552 | 24752 |
-10 | -70 | -350 | -1540 | -6370 | -25480 | -99960 |
28 | 224 | 1232 | 5824 | 25480 | 106624 | 434112 |
-84 | -756 | -4536 | -22932 | -105840 | -462672 | -1953504 |
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