Generating function
$$U_{1296}(x, y) = \sqrt{\frac{x \left(1 - \sqrt{1 - 4 y}\right)^{3}}{2 y^{3}} + 1}$$
Explicit formula
$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{1296}(n, m, k) = \frac{3 n \operatorname{Tsqrt}{\left(n,k \right)} {\binom{2 m + 3 n - 1}{m}}}{m + 3 n}$$
Data table
nan 0 0 0 0 0 0
2 6 18 56 18 594 2002
-2 -12 -54 -220 -858 -3276 -12376
4 36 216 1092 5040 22032 93024
-10 -120 -900 -5440 -29070 -143640 -672980
28 420 3780 26600 161700 892584 4604600
-84 -1512 -15876 -127512 -869400 -5307120 -29926260
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