Av(1243, 2314, 3124)
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Generating Function
\(\displaystyle \frac{2 x^{2}-2 x +1-\sqrt{-4 x^{5}+16 x^{4}-24 x^{3}+20 x^{2}-8 x +1}}{2 \left(x^{2}-x +1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 21, 77, 290, 1118, 4398, 17595, 71385, 293042, 1215035, 5081259, 21408350, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-x +1\right) x F \left(x \right)^{2}+\left(-2 x^{2}+2 x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 290\)
\(\displaystyle a \! \left(n +7\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(17+10 n \right) a \! \left(1+n \right)}{n +8}+\frac{2 \left(59+22 n \right) a \! \left(n +2\right)}{n +8}-\frac{12 \left(18+5 n \right) a \! \left(n +3\right)}{n +8}+\frac{4 \left(59+13 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(160+29 n \right) a \! \left(n +5\right)}{n +8}+\frac{3 \left(20+3 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7\)
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This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 29 rules.

Found on July 23, 2021.

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