Av(1324, 1342, 1423)
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Counting Sequence
1, 1, 2, 6, 21, 79, 310, 1251, 5151, 21536, 91137, 389510, 1678565, 7284975, 31811311, ...
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Implicit Equation for the Generating Function
\(\displaystyle x F \left(x \right)^{4}+\left(-x -1\right) F \left(x \right)^{3}+\left(x^{2}-2 x +3\right) F \left(x \right)^{2}+\left(2 x -3\right) F \! \left(x \right)+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 79\)
\(\displaystyle a \! \left(6\right) = 310\)
\(\displaystyle a \! \left(7\right) = 1251\)
\(\displaystyle a \! \left(8\right) = 5151\)
\(\displaystyle a \! \left(n +9\right) = \frac{n \left(2 n +3\right) \left(2 n -1\right) a \! \left(n \right)}{\left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(8 n^{3}+83 n^{2}+145 n +55\right) a \! \left(n +1\right)}{\left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(1151 n^{3}+8520 n^{2}+22579 n +18546\right) a \! \left(n +2\right)}{12 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}-\frac{\left(2233 n^{3}+55122 n^{2}+279635 n +395634\right) a \! \left(n +3\right)}{24 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(62879 n^{3}+790140 n^{2}+3299443 n +4570050\right) a \! \left(n +4\right)}{24 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}-\frac{\left(67265 n^{3}+1013004 n^{2}+5056489 n +8372154\right) a \! \left(n +5\right)}{24 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(7612 n^{3}+136431 n^{2}+810569 n +1597989\right) a \! \left(n +6\right)}{6 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}-\frac{\left(3640 n^{3}+76029 n^{2}+526883 n +1212402\right) a \! \left(n +7\right)}{12 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(458 n^{3}+10893 n^{2}+85993 n +225450\right) a \! \left(n +8\right)}{12 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}, \quad n \geq 9\)
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This specification was found using the strategy pack "Insertion Point Placements Tracked Fusion" and has 42 rules.

Found on July 23, 2021.

Finding the specification took 9 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_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\ F_{20}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{6}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{14}\! \left(x \right)+F_{31}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= -\frac{y \left(F_{21}\! \left(x , 1\right)-F_{21}\! \left(x , y\right)\right)}{-1+y}\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{15}\! \left(x , y\right) F_{6}\! \left(x \right)\\ F_{38}\! \left(x , y\right) &= -\frac{y \left(F_{27}\! \left(x , 1\right)-F_{27}\! \left(x , y\right)\right)}{-1+y}\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{14}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{26}\! \left(x , 1\right)\\ \end{align*}\)