Av(1243, 1342, 2341, 3124)
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Generating Function
\(\displaystyle \frac{\left(-x^{3}+2 x^{2}-2 x +1\right) \sqrt{1-4 x}+x^{3}+2 x -1}{2 x \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 20, 66, 215, 702, 2321, 7800, 26642, 92335, 324049, 1149436, 4114374, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x \right)^{2}-\left(x^{3}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{6}-4 x^{5}+9 x^{4}-11 x^{3}+10 x^{2}-5 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 215\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{\left(10+9 n \right) a \! \left(n +1\right)}{n +5}-\frac{\left(10 n +29\right) a \! \left(n +2\right)}{n +5}+\frac{\left(23+6 n \right) a \! \left(n +3\right)}{n +5}+\frac{\frac{3}{2} n^{2}+\frac{9}{2} n +12}{n +5}, \quad n \geq 7\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 96 rules.

Found on July 23, 2021.

Finding the specification took 5 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_{15}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{15}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \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_{15}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= x\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{15}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{21}\! \left(x \right) &= 0\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{15}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{15}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{15}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{15}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{15}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{50}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{15}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{15}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x , 1\right)\\ F_{62}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{64}\! \left(x , y\right)+F_{66}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{63}\! \left(x , y\right) F_{65}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= y x\\ F_{66}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{67}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= \frac{F_{63}\! \left(x , y\right) y -F_{63}\! \left(x , 1\right)}{-1+y}\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{71}\! \left(x \right)+F_{79}\! \left(x , y\right)+F_{85}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\ F_{71}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{15}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{15}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{79}\! \left(x , y\right) &= F_{65}\! \left(x , y\right) F_{80}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= \frac{F_{81}\! \left(x , y\right) y -F_{81}\! \left(x , 1\right)}{-1+y}\\ F_{81}\! \left(x , y\right) &= F_{71}\! \left(x \right)+F_{82}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{65}\! \left(x , y\right) F_{81}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{81}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{86}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= \frac{F_{87}\! \left(x , y\right) y -F_{87}\! \left(x , 1\right)}{-1+y}\\ F_{87}\! \left(x , y\right) &= F_{71}\! \left(x \right)+F_{88}\! \left(x , y\right)+F_{92}\! \left(x , y\right)+F_{93}\! \left(x \right)\\ F_{88}\! \left(x , y\right) &= F_{89}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right) F_{65}\! \left(x , y\right) F_{90}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{91}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{65}\! \left(x , y\right) F_{90}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{87}\! \left(x , y\right)\\ F_{93}\! \left(x \right) &= F_{83}\! \left(x , 1\right)\\ F_{94}\! \left(x , y\right) &= F_{95}\! \left(x , y\right)\\ F_{95}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{80}\! \left(x , y\right)\\ \end{align*}\)