Av(1243, 1324, 1342, 2314, 2341, 4123)
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Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{3} \sqrt{-4 x +1}-2 x^{5}-2 x^{4}+x^{3}-3 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 18, 51, 148, 454, 1466, 4911, 16860, 58867, 208112, 743021, 2674584, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x \right)^{2}+\left(x^{2}+1\right) \left(2 x^{3}+2 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}+2 x^{8}+3 x^{6}-6 x^{5}+13 x^{4}-19 x^{3}+15 x^{2}-6 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 148\)
\(\displaystyle a \! \left(n +1\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +2}-\frac{3 n^{3}-14 n^{2}+11 n +6}{n +2}, \quad n \geq 7\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 33 rules.

Found on January 20, 2022.

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