Av(13452, 14253, 14352, 23451, 24153, 24351, 34152, 34251)
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Generating Function
\(\displaystyle \frac{4 x -5+\sqrt{8 x^{2}-8 x +1}}{4 x -4}\)
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Counting Sequence
1, 1, 2, 6, 24, 112, 568, 3032, 16768, 95200, 551616, 3248704, 19389824, 117021824, 712934784, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -2\right) F \left(x \right)^{2}+\left(-4 x +5\right) F \! \left(x \right)+x -3 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 n a \! \left(n \right)}{n +3}-\frac{4 \left(3+4 n \right) a \! \left(n +1\right)}{n +3}+\frac{3 \left(5+3 n \right) a \! \left(n +2\right)}{n +3}, \quad n \geq 3\)
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This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 22 rules.

Found on January 23, 2022.

Finding the specification took 59 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= y x\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{12}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{19}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{12}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 30 rules.

Found on January 22, 2022.

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