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
$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{1293}(n, m, k) = \frac{\left(3 k + n\right) \operatorname{Tsqrt}{\left(n,k \right)} {\binom{3 k + 2 m + n - 1}{m}}}{3 k + m + n}$$
Data table
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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