Generating function
$$U_{933}(x, y) = - \frac{1}{2 x y + x \sqrt{4 y^{2} + 1} - 1}$$
Explicit formula
$$Tsqrt2(n, k) = \begin{cases}\frac {k \binom{n+j}{n} \binom{2n+2j}{n+j}} {\binom{2j}{j} {n+k}} &\text{if n+k even, where j =} \frac{n+k} {2},\\\frac {k \binom{n+j}{n} \binom{2n+2j}{n+j}} {\binom{2j}{j} {n+k}} &\text{if n+k odd, where j =} \frac{n+k+1} {2},\\\end{cases} $$$$T_{933}(n, m, k) = \begin{cases}1&\text{if n = 0,m = 0} ,\ \\\frac{k \operatorname{Tsqrt_{2}}{\left(m,n \right)} {\binom{k + n}{n}}}{k + n} \end{cases} $$
1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 2 | 2 | 0 | -2 | 0 | 4 |
1 | 4 | 8 | 8 | 0 | -8 | 0 |
1 | 6 | 18 | 32 | 30 | 0 | -28 |
1 | 8 | 32 | 80 | 128 | 112 | 0 |
1 | 10 | 50 | 160 | 350 | 512 | 420 |
1 | 12 | 72 | 280 | 768 | 1512 | 2048 |
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