Av(1342, 13254)
Counting Sequence
1, 1, 2, 6, 23, 102, 495, 2549, 13682, 75714, 428882, 2474573, 14492346, 85926361, 514763279, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(x^{2}-x +1\right) F \left(x \right)^{3}+\left(x -3\right) F \left(x \right)^{2}+3 F \! \left(x \right)-1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(n +4\right) = \frac{24 \left(3 n +4\right) \left(3 n +2\right) a \! \left(n \right)}{\left(2 n +7\right) \left(n +3\right)}-\frac{\left(469 n^{2}+953 n +378\right) a \! \left(n +1\right)}{2 \left(2 n +7\right) \left(n +3\right)}+\frac{3 \left(155 n^{2}+309 n +112\right) a \! \left(n +2\right)}{2 \left(2 n +7\right) \left(n +3\right)}-\frac{3 \left(11 n^{2}+21 n -16\right) a \! \left(n +3\right)}{2 \left(2 n +7\right) \left(n +3\right)}, \quad n \geq 4$$

This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Req Corrob Expand Verified" and has 22 rules.

Found on January 22, 2022.

Finding the specification took 62 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_{19}\! \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_{0}\! \left(x \right) F_{19}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)^{2} F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{19}\! \left(x \right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x , y\right)\\ F_{19}\! \left(x \right) &= x\\ F_{20}\! \left(x , y\right) &= F_{19}\! \left(x \right) F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\ \end{align*}

This specification was found using the strategy pack "Point Placements Tracked Fusion Expand Verified" and has 24 rules.

Found on January 22, 2022.

Finding the specification took 120 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_{16}\! \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_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)^{2} F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{16}\! \left(x \right) F_{8}\! \left(x , y\right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{7}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= -\frac{y \left(F_{23}\! \left(x , 1\right)-F_{23}\! \left(x , y\right)\right)}{-1+y}\\ F_{8}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ \end{align*}