Av(1342, 2143, 4132)
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Generating Function
\(\displaystyle \frac{\left(2 x^{3}-3 x^{2}+3 x -1\right) \sqrt{1-4 x}+4 x^{4}-14 x^{3}+17 x^{2}-7 x +1}{8 x^{4}-16 x^{3}+12 x^{2}-2 x}\)
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Counting Sequence
1, 1, 2, 6, 21, 75, 268, 961, 3467, 12591, 46012, 169088, 624478, 2316582, 8627816, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(4 x^{3}-8 x^{2}+6 x -1\right) F \left(x \right)^{2}+\left(-4 x^{4}+14 x^{3}-17 x^{2}+7 x -1\right) F \! \left(x \right)+x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 268\)
\(\displaystyle a \! \left(7\right) = 961\)
\(\displaystyle a \! \left(n +7\right) = \frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +8}-\frac{24 \left(5 n +7\right) a \! \left(1+n \right)}{n +8}+\frac{20 \left(11 n +28\right) a \! \left(n +2\right)}{n +8}-\frac{48 \left(5 n +19\right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(82 n +407\right) a \! \left(n +4\right)}{n +8}-\frac{\left(65 n +394\right) a \! \left(n +5\right)}{n +8}+\frac{\left(92+13 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 8\)
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This specification was found using the strategy pack "Row And Col Placements" and has 31 rules.

Found on July 23, 2021.

Finding the specification took 15 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{3}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{21}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{3}\! \left(x \right) F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right) F_{3}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{26}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{26}\! \left(x \right) F_{5}\! \left(x \right)\\ \end{align*}\)