Av(1234, 1243, 1342, 2341, 4123)
Generating Function
\(\displaystyle \frac{-2 \left(x -\frac{1}{2}\right) \left(x +1\right) \left(x -1\right)^{3} \sqrt{1-4 x}+8 x^{5}-15 x^{4}+10 x^{3}+2 x^{2}-4 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 60, 189, 602, 1957, 6510, 22123, 76549, 268778, 954928, 3425439, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \left(x
\right)^{2}-\left(2 x -1\right) \left(8 x^{5}-15 x^{4}+10 x^{3}+2 x^{2}-4 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+4 x^{10}-5 x^{9}-22 x^{8}+84 x^{7}-118 x^{6}+67 x^{5}+19 x^{4}-52 x^{3}+32 x^{2}-9 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 189\)
\(\displaystyle a \! \left(7\right) = 602\)
\(\displaystyle a \! \left(8\right) = 1957\)
\(\displaystyle a \! \left(n +4\right) = \frac{8 \left(-1+2 n \right) a \! \left(n \right)}{n +5}-\frac{4 \left(n -3\right) a \! \left(n +1\right)}{n +5}-\frac{2 \left(19+6 n \right) a \! \left(n +2\right)}{n +5}+\frac{\left(27+7 n \right) a \! \left(n +3\right)}{n +5}+\frac{3 n^{3}-20 n^{2}+18 n -11}{n +5}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 189\)
\(\displaystyle a \! \left(7\right) = 602\)
\(\displaystyle a \! \left(8\right) = 1957\)
\(\displaystyle a \! \left(n +4\right) = \frac{8 \left(-1+2 n \right) a \! \left(n \right)}{n +5}-\frac{4 \left(n -3\right) a \! \left(n +1\right)}{n +5}-\frac{2 \left(19+6 n \right) a \! \left(n +2\right)}{n +5}+\frac{\left(27+7 n \right) a \! \left(n +3\right)}{n +5}+\frac{3 n^{3}-20 n^{2}+18 n -11}{n +5}, \quad n \geq 9\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 139 rules.
Found on July 23, 2021.Finding the specification took 4 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_{14}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{44}\! \left(x , y\right)+F_{46}\! \left(x \right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{11}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= y x\\
F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x , y\right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{23}\! \left(x \right) &= 0\\
F_{24}\! \left(x \right) &= F_{14}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{23}\! \left(x \right)+F_{33}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{19}\! \left(x \right)+F_{32}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{28}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{39}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= 2 F_{23}\! \left(x \right)+F_{41}\! \left(x , y\right)+F_{43}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{36}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x , 1\right)\\
F_{47}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{19}\! \left(x \right)+F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{23}\! \left(x \right)+F_{51}\! \left(x , y\right)+F_{53}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{52}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\
F_{54}\! \left(x \right) &= F_{14}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{14}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{14}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{14}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{14}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{14}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{14}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{14}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{14}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{137}\! \left(x \right)+F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{89}\! \left(x \right) &= F_{14}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x , 1\right)\\
F_{91}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x , y\right)+F_{92}\! \left(x , y\right)+F_{94}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{93}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\
F_{94}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{95}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= -\frac{-y F_{91}\! \left(x , y\right)+F_{91}\! \left(x , 1\right)}{-1+y}\\
F_{96}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{97}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)+F_{105}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{102}\! \left(x \right) &= 0\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{110}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{105}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)+F_{122}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{118}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= F_{116}\! \left(x , y\right)\\
F_{116}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{117}\! \left(x , y\right)\\
F_{117}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{19}\! \left(x \right)\\
F_{118}\! \left(x , y\right) &= 2 F_{23}\! \left(x \right)+F_{119}\! \left(x , y\right)+F_{121}\! \left(x , y\right)\\
F_{119}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{120}\! \left(x , y\right)\\
F_{120}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{22}\! \left(x \right)\\
F_{121}\! \left(x , y\right) &= F_{114}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{122}\! \left(x , y\right) &= F_{123}\! \left(x , y\right)+F_{126}\! \left(x , y\right)\\
F_{123}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)\\
F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{125}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{123}\! \left(x , y\right)\\
F_{126}\! \left(x , y\right) &= 3 F_{23}\! \left(x \right)+F_{127}\! \left(x , y\right)+F_{129}\! \left(x , y\right)\\
F_{127}\! \left(x , y\right) &= F_{128}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{128}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{126}\! \left(x , y\right)\\
F_{129}\! \left(x , y\right) &= F_{130}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)+F_{134}\! \left(x , y\right)\\
F_{131}\! \left(x , y\right) &= F_{132}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{133}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{133}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{131}\! \left(x , y\right)\\
F_{134}\! \left(x , y\right) &= 3 F_{23}\! \left(x \right)+F_{129}\! \left(x , y\right)+F_{135}\! \left(x , y\right)\\
F_{135}\! \left(x , y\right) &= F_{136}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{136}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{134}\! \left(x , y\right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{99}\! \left(x \right)\\
\end{align*}\)