Generating function
$$U_{1361}(x, y) = \frac{\left(1 - \sqrt{1 - 4 y}\right)^{2} \sqrt{\frac{x \left(1 - \sqrt{1 - 4 y}\right)^{3}}{2 y^{3}} + 1}}{4 y^{2}}$$
Explicit formula
$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{1361}(n, m, k) = \frac{\left(2 k + 3 n\right) \operatorname{Tsqrt}{\left(n,k \right)} {\binom{2 k + 2 m + 3 n - 1}{m}}}{2 k + m + 3 n}$$
Data table
1 2 5 14 42 132 429
2 1 4 15 55 2002 728
-2 -16 -88 -416 -1820 -7616 -31008
4 44 308 1760 8976 42636 193116
-10 -140 -1190 -7980 -46550 -247940 -1239700
28 476 4760 36652 240856 1423240 7796880
-84 -1680 -19320 -168000 -1228500 -7980336 -47575080
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