Av(1324, 1342, 2341)
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Generating Function
\(\displaystyle -\frac{\left(\sqrt{-4 x +1}+1\right) \left(-1+3 x \right)}{\sqrt{-4 x +1}\, \left(2 x^{2}-6 x +2\right)}\)
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Counting Sequence
1, 1, 2, 6, 21, 78, 297, 1143, 4419, 17119, 66386, 257621, 1000407, 3887666, 15119991, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x -1\right) \left(x^{2}-3 x +1\right)^{2} F \left(x \right)^{2}+\left(4 x -1\right) \left(-1+3 x \right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x \left(-1+3 x \right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{6 \left(3+2 n \right) a \! \left(n \right)}{n +4}+\frac{\left(74+43 n \right) a \! \left(n +1\right)}{n +4}-\frac{2 \left(41+17 n \right) a \! \left(n +2\right)}{n +4}+\frac{2 \left(16+5 n \right) a \! \left(n +3\right)}{n +4}, \quad n \geq 4\)
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This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 34 rules.

Found on July 23, 2021.

Finding the specification took 4 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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= \frac{y F_{5}\! \left(x , y\right)-F_{5}\! \left(x , 1\right)}{-1+y}\\ F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{11}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= y x\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{5}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x \right) F_{21}\! \left(x , y\right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{18}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{30}\! \left(x \right) &= 0\\ F_{31}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\ \end{align*}\)