Av(12453, 12543, 14253, 14523, 14532, 15243, 15423, 15432, 21453, 21543)
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Counting Sequence
1, 1, 2, 6, 24, 110, 542, 2800, 14966, 82074, 459208, 2610938, 15042218, 87621664, 515190026, ...
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Implicit Equation for the Generating Function
\(\displaystyle -F \left(x \right)^{3}+\left(x +2\right) F \left(x \right)^{2}-3 x F \! \left(x \right)+x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 110\)
\(\displaystyle a \! \left(n +6\right) = \frac{5 \left(n -1\right) n a \! \left(n \right)}{7 \left(n +6\right) \left(n +5\right)}-\frac{3 n \left(15 n +11\right) a \! \left(n +1\right)}{7 \left(n +6\right) \left(n +5\right)}+\frac{6 \left(71 n +120\right) \left(n +1\right) a \! \left(n +2\right)}{35 \left(n +6\right) \left(n +5\right)}-\frac{\left(197 n^{2}+301 n -606\right) a \! \left(n +3\right)}{35 \left(n +6\right) \left(n +5\right)}+\frac{6 \left(3 n^{2}-55 n -254\right) a \! \left(n +4\right)}{35 \left(n +6\right) \left(n +5\right)}+\frac{3 \left(11 n +52\right) a \! \left(n +5\right)}{5 \left(n +6\right)}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 44 rules.

Found on January 25, 2022.

Finding the specification took 4457 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_{20}\! \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_{20}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{20}\! \left(x \right)}\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{20}\! \left(x \right)}\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{20}\! \left(x \right) &= x\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{20}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{20}\! \left(x \right)}\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= -F_{39}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\ F_{29}\! \left(x \right) &= -F_{37}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= -F_{34}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{20}\! \left(x \right)}\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{18}\! \left(x \right) F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{20}\! \left(x \right) F_{23}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\ \end{align*}\)