Av(1243, 1342, 2341, 4123)
Generating Function
\(\displaystyle \frac{2 \left(x -\frac{1}{2}\right) \left(x -1\right)^{3} \sqrt{1-4 x}-6 x^{5}+8 x^{4}-x^{3}-7 x^{2}+5 x -1}{2 x \left(2 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 213, 683, 2211, 7291, 24552, 84305, 294297, 1041213, 3723752, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{8} F \left(x
\right)^{2}+\left(2 x -1\right) \left(6 x^{5}-8 x^{4}+x^{3}+7 x^{2}-5 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+9 x^{9}-20 x^{8}-10 x^{7}+109 x^{6}-210 x^{5}+218 x^{4}-137 x^{3}+52 x^{2}-11 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 213\)
\(\displaystyle a \! \left(7\right) = 683\)
\(\displaystyle a \! \left(n +4\right) = -\frac{8 \left(2 n +3\right) a \! \left(n \right)}{5+n}+\frac{4 \left(9 n +19\right) a \! \left(1+n \right)}{5+n}-\frac{2 \left(14 n +41\right) a \! \left(n +2\right)}{5+n}+\frac{\left(9 n +35\right) a \! \left(n +3\right)}{5+n}+\frac{3 n^{3}-14 n^{2}-9 n +4}{2 n +10}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 213\)
\(\displaystyle a \! \left(7\right) = 683\)
\(\displaystyle a \! \left(n +4\right) = -\frac{8 \left(2 n +3\right) a \! \left(n \right)}{5+n}+\frac{4 \left(9 n +19\right) a \! \left(1+n \right)}{5+n}-\frac{2 \left(14 n +41\right) a \! \left(n +2\right)}{5+n}+\frac{\left(9 n +35\right) a \! \left(n +3\right)}{5+n}+\frac{3 n^{3}-14 n^{2}-9 n +4}{2 n +10}, \quad n \geq 8\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 138 rules.
Found on July 23, 2021.Finding the specification took 6 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{42}\! \left(x \right)+F_{6}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= y x\\
F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{11}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{12}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{13}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x , y\right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \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_{12}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{21}\! \left(x \right)+F_{31}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{30}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{26}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= 2 F_{21}\! \left(x \right)+F_{39}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{40}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{34}\! \left(x , y\right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x , 1\right)\\
F_{44}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x , y\right)+F_{47}\! \left(x , y\right)+F_{49}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{47}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= -\frac{-y F_{44}\! \left(x , y\right)+F_{44}\! \left(x , 1\right)}{-1+y}\\
F_{49}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{52}\! \left(x \right)+F_{66}\! \left(x , y\right)\\
F_{52}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{55}\! \left(x \right) &= 0\\
F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{58}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{68}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= 2 F_{21}\! \left(x \right)+F_{72}\! \left(x , y\right)+F_{74}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{20}\! \left(x \right)+F_{71}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{67}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{78}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{76}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= 3 F_{21}\! \left(x \right)+F_{80}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{83}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{86}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= 3 F_{21}\! \left(x \right)+F_{82}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{89}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{90}\! \left(x \right) &= F_{12}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{12}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{106}\! \left(x \right)+F_{92}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{12}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{104}\! \left(x \right)+F_{6}\! \left(x \right)+F_{90}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{12}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{100}\! \left(x \right)+F_{104}\! \left(x \right)+F_{105}\! \left(x \right)+F_{6}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{12}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x , 1\right)\\
F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{102}\! \left(x , y\right) &= -\frac{-y F_{103}\! \left(x , y\right)+F_{103}\! \left(x , 1\right)}{-1+y}\\
F_{103}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{101}\! \left(x , y\right)+F_{104}\! \left(x \right)+F_{45}\! \left(x , y\right)+F_{49}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{104}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{49}\! \left(x , 1\right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{118}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{125}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{12}\! \left(x \right) F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{12}\! \left(x \right) F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{12}\! \left(x \right) F_{130}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{132}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{133}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{12}\! \left(x \right) F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{12}\! \left(x \right) F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{20}\! \left(x \right)\\
\end{align*}\)