Generating function
$$U_{1306}(x, y) = \frac{2 y^{3} \left(- \frac{x \left(1 - \sqrt{1 - 4 y}\right)}{y} - \sqrt{- \frac{2 x \left(1 - \sqrt{1 - 4 y}\right)}{y} + 1} + 1\right)^{2}}{x^{4} \left(1 - \sqrt{1 - 4 y}\right)^{3}}$$
Explicit formula
$$T_{1306}(n, m, k) = \frac{2 k \left(k + n\right) {\binom{4 k + 2 n}{n}} {\binom{k + 2 m + n - 1}{m}}}{\left(2 k + n\right) \left(k + m + n\right)}$$
Data table
1 1 2 5 14 42 132
4 8 2 56 168 528 1716
14 42 126 392 126 4158 14014
48 192 672 2304 792 27456 96096
165 825 33 12375 45375 165165 6006
572 3432 15444 6292 245388 936936 3539536
2002 14014 7007 308308 1275274 5101096 20011992
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