Av(1234, 1324, 1342, 3124)
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Generating Function
\(\displaystyle \frac{-x \sqrt{-4 x +1}-12 x^{2}+11 x -2}{2 \left(4 x -1\right) \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 20, 69, 244, 881, 3234, 12022, 45120, 170596, 648788, 2479058, 9509628, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x -1\right) \left(x -1\right)^{4} F \left(x \right)^{2}+\left(4 x -1\right) \left(3 x -2\right) \left(x -1\right)^{2} F \! \left(x \right)+9 x^{3}-14 x^{2}+7 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +2\right) = -\frac{2 \left(2 n +3\right) a \! \left(n \right)}{n +1}+\frac{\left(4+5 n \right) a \! \left(n +1\right)}{n +1}+\frac{4}{n +1}, \quad n \geq 3\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 28 rules.

Found on July 23, 2021.

Finding the specification took 39 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_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ 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_{4}\! \left(x \right)\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x , y\right) &= F_{1}\! \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_{14}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\ F_{24}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x , y\right) &= -\frac{-y F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)