Generating function
$$U_{1076}(x, y) = 2 x \left(y + 1\right)^{4} + \left(y + 1\right)^{2} \sqrt{4 x^{2} \left(y + 1\right)^{4} + 1}$$
Explicit formula
$$Tsqrt2(n, k) = \begin{cases}\frac {k \binom{n+j}{n} \binom{2n+2j}{n+j}} {\binom{2j}{j} {n+k}} &\text{if n+k even, where j =} \frac{n+k} {2},\\\frac {k \binom{n+j}{n} \binom{2n+2j}{n+j}} {\binom{2j}{j} {n+k}} &\text{if n+k odd, where j =} \frac{n+k+1} {2},\\\end{cases} $$$$T_{1076}(n, m, k) = k \operatorname{Tsqrt_{2}}{\left(n,k \right)} {\binom{2 k + 2 n}{m}}$$
Data table
1 2 1 0 0 0 0
2 8 12 8 2 0 0
2 12 3 4 3 12 2
0 0 0 0 0 0 0
-2 -20 -90 -240 -420 -504 -420
0 0 0 0 0 0 0
4 56 364 1456 4004 8008 12012
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