Av(13452, 13542, 14352, 14532, 15342, 15432, 31452, 31542)
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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 101 rules.

Found on January 23, 2022.

Finding the specification took 264 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_{100}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= y x\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\ F_{25}\! \left(x \right) &= 0\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{100}\! \left(x \right) F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{49}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\ F_{49}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{52}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{57}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{61}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{67}\! \left(x , y\right)+F_{80}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{72}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{51}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{77}\! \left(x , y\right)+F_{78}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\ F_{77}\! \left(x , y\right) &= F_{60}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{79}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= 0\\ F_{81}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{82}\! \left(x , y\right)+F_{83}\! \left(x , y\right)+F_{86}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{70}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{87}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= 0\\ F_{89}\! \left(x , y\right) &= F_{9}\! \left(x , y\right) F_{90}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{92}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{78}\! \left(x , y\right)+F_{80}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\ F_{93}\! \left(x , y\right) &= F_{9}\! \left(x , y\right) F_{94}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)\\ F_{95}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{83}\! \left(x , y\right)+F_{88}\! \left(x , y\right)+F_{96}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{9}\! \left(x , y\right) F_{97}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{9}\! \left(x , y\right) F_{99}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{91}\! \left(x , y\right)\\ F_{100}\! \left(x \right) &= x\\ \end{align*}\)