Av(1243, 1324, 1342, 1423, 2314, 2341)
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Generating Function
\(\displaystyle \frac{\left(x -1\right)^{2} \sqrt{-4 x +1}+2 x^{3}-x^{2}+2 x -1}{4 x^{2}-2 x}\)
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Counting Sequence
1, 1, 2, 6, 18, 55, 172, 551, 1806, 6043, 20588, 71232, 249700, 885062, 3166776, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} F \left(x \right)^{2}-\left(x^{2}+1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{5}-2 x^{3}+5 x^{2}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(n +3\right) = \frac{4 \left(-1+2 n \right) a \! \left(n \right)}{n +4}-\frac{2 \left(10+7 n \right) a \! \left(1+n \right)}{n +4}+\frac{\left(19+7 n \right) a \! \left(n +2\right)}{n +4}, \quad n \geq 5\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 33 rules.

Found on July 23, 2021.

Finding the specification took 7 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_{22}\! \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_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \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_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \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_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x , 1\right)\\ F_{25}\! \left(x , y\right) &= \frac{y F_{26}\! \left(x , y\right)-F_{26}\! \left(x , 1\right)}{-1+y}\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= y x\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{25}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\ \end{align*}\)