Av(1243, 1324, 1342, 1423, 2314)
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Generating Function
\(\displaystyle \frac{\left(x -\frac{1}{2}\right) \left(x^{2}+2 x -1+\left(x -1\right)^{2} \sqrt{-4 x +1}\right)}{\left(x^{4}-4 x^{3}+8 x^{2}-5 x +1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 19, 61, 199, 660, 2223, 7593, 26262, 91850, 324427, 1155980, 4150950, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-4 x^{3}+8 x^{2}-5 x +1\right) x F \left(x \right)^{2}-\left(2 x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+\left(2 x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 199\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(52+23 n \right) a \! \left(1+n \right)}{n +8}+\frac{\left(363+127 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(688+181 n \right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(334+69 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(335+57 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(83+12 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 29 rules.

Found on July 23, 2021.

Finding the specification took 2 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{0}\! \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_{12}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\ F_{24}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{27}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= y x\\ F_{27}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= \frac{y F_{24}\! \left(x , y\right)-F_{24}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)