Generating function
$$U_{1198}(x, y) = \frac{\left(1 - \sqrt{1 - 4 y}\right)^{3} \sqrt{4 x + 1}}{8 y^{3}}$$
Explicit formula
$$Tsqrt(n, k) = \begin{cases}\binom{m}{n} 4^{n}&\text{if k even, where m =} \frac{k} {2},\\\frac {{(-1)}^{n-m} \binom{n}{m} \binom{2n}{n}} {\binom{2n}{2m}} &\text{if k odd, n > m, where m =} \frac{k+1} {2},\\\frac {\binom{2m}{2n} \binom{2n}{n}} {\binom{m}{n}}&\text{if k odd, n} \le \text{m, where m =} \frac{k+1} {2},\\\end{cases} $$$$T_{1198}(n, m, k) = \frac{3 k \operatorname{Tsqrt}{\left(n,k \right)} {\binom{3 k + 2 m - 1}{m}}}{k + m}$$
3 | 6 | 15 | 42 | 126 | 396 | 1287 |
6 | 12 | 3 | 84 | 252 | 792 | 2574 |
-6 | -12 | -30 | -84 | -252 | -792 | -2574 |
12 | 24 | 60 | 168 | 504 | 1584 | 5148 |
-30 | -60 | -150 | -420 | -1260 | -3960 | -12870 |
84 | 168 | 420 | 1176 | 3528 | 11088 | 36036 |
-252 | -504 | -1260 | -3528 | -10584 | -33264 | -108108 |
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