Generating function
$$U_{1360}(x, y) = \left(y + 1\right)^{3} \sqrt{4 x \left(y + 1\right)^{2} + 1}$$
Explicit formula
$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{1360}(n, m, k) = \operatorname{Tsqrt}{\left(n,k \right)} {\binom{3 k + 2 n}{m}}$$
Data table
1 3 3 1 0 0 0
2 1 2 2 1 2 0
-2 -14 -42 -70 -70 -42 -14
4 36 144 336 504 504 336
-10 -110 -550 -1650 -3300 -4620 -4620
28 364 2184 8008 20020 36036 48048
-84 -1260 -8820 -38220 -114660 -252252 -420420
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