Av(12453, 12543, 14253, 14523, 15243, 15423, 21453, 21543)
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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 "Point Placements Tracked Fusion" and has 40 rules.

Found on January 25, 2022.

Finding the specification took 1766 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_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{13}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= \frac{F_{18}\! \left(x \right)}{F_{0}\! \left(x \right)}\\ F_{18}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= -F_{25}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{14}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{30}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)