Av(1324, 1342, 2341, 3124)
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Generating Function
\(\displaystyle \frac{-\left(-1+x \right)^{2} \sqrt{-4 x +1}-x^{2}-2 x +1}{4 x^{3}-6 x^{2}+2 x}\)
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Counting Sequence
1, 1, 2, 6, 20, 67, 223, 742, 2484, 8399, 28731, 99451, 348127, 1231141, 4393821, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(-1+x \right)^{2} F \left(x \right)^{2}+\left(2 x -1\right) \left(-1+x \right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{4+n}-\frac{2 \left(13+7 n \right) a \! \left(1+n \right)}{4+n}+\frac{\left(20+7 n \right) a \! \left(n +2\right)}{4+n}+\frac{6}{4+n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 26 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_{3}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x , y\right) &= \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{17}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{14}\! \left(x , y\right) &= \frac{y F_{12}\! \left(x , y\right)-F_{12}\! \left(x , 1\right)}{-1+y}\\ F_{15}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= y x\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{24}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{24}\! \left(x , y\right) &= \frac{y F_{12}\! \left(x , y\right)-F_{12}\! \left(x , 1\right)}{-1+y}\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ \end{align*}\)