###### Av(1324, 1342, 2431)
Generating Function
$$\displaystyle \frac{\left(-5 x^{3}+8 x^{2}-5 x +1\right) \sqrt{1-4 x}-2 x^{4}+13 x^{3}-16 x^{2}+7 x -1}{4 x^{4}-2 x^{3}}$$
Counting Sequence
1, 1, 2, 6, 21, 77, 285, 1054, 3889, 14330, 52800, 194748, 719602, 2664989, 9894443, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(2 x -1\right)^{2} x^{3} F \left(x \right)^{2}+\left(x^{3}-6 x^{2}+5 x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{5}+12 x^{4}-28 x^{3}+23 x^{2}-8 x +1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(4\right) = 21$$
$$\displaystyle a \! \left(5\right) = 77$$
$$\displaystyle a \! \left(n +5\right) = \frac{20 \left(1+2 n \right) a \! \left(n \right)}{8+n}-\frac{2 \left(117+23 n \right) a \! \left(3+n \right)}{8+n}-\frac{2 \left(95+47 n \right) a \! \left(n +1\right)}{8+n}+\frac{\left(334+93 n \right) a \! \left(n +2\right)}{8+n}+\frac{\left(72+11 n \right) a \! \left(n +4\right)}{8+n}, \quad n \geq 6$$

### This specification was found using the strategy pack "Point Placements" and has 22 rules.

Found on July 23, 2021.

Finding the specification took 10 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_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{1}\! \left(x \right)\\ \end{align*}