Generating function
$$U_{1298}(x, y) = \frac{x \left(1 - \sqrt{1 - 4 y}\right)}{y} + \sqrt{\frac{x^{2} \left(1 - \sqrt{1 - 4 y}\right)^{2}}{y^{2}} + 1}$$
Explicit formula
$$Tsqrt2(n, k) = \begin{cases}\frac {k \binom{n+j}{n} \binom{2n+2j}{n+j}} {\binom{2j}{j} {n+k}} &\text{if n+k even, where j =} \frac{n+k} {2},\\\frac {k \binom{n+j}{n} \binom{2n+2j}{n+j}} {\binom{2j}{j} {n+k}} &\text{if n+k odd, where j =} \frac{n+k+1} {2},\\\end{cases} $$$$T_{1298}(n, m, k) = \begin{cases}1&\text{if m = 0,n = 0} ,\ \\\frac{n \operatorname{Tsqrt_{2}}{\left(n,k \right)} {\binom{2 m + n - 1}{m}}}{m + n} \end{cases} $$
Data table
1 0 0 0 0 0 0
2 2 4 1 28 84 264
2 4 1 28 84 264 858
0 0 0 0 0 0 0
-2 -8 -28 -96 -330 -1144 -4004
0 0 0 0 0 0 0
4 24 108 440 1716 6552 24752
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