Av(1234, 1243, 1324, 2134, 2341, 4123)
Generating Function
\(\displaystyle \frac{-\left(x^{2}+x -1\right) \left(x -1\right)^{4} \sqrt{1-4 x}-4 x^{8}+8 x^{7}-x^{6}-x^{5}-x^{4}+6 x^{3}-9 x^{2}+5 x -1}{2 x \left(x^{2}+x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 18, 54, 158, 478, 1514, 4998, 17008, 59109, 208498, 743628, 2675532, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{8} F \left(x
\right)^{2}+\left(x^{2}+x -1\right) \left(4 x^{8}-8 x^{7}+x^{6}+x^{5}+x^{4}-6 x^{3}+9 x^{2}-5 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{15}-16 x^{14}+18 x^{13}-x^{12}-8 x^{11}-3 x^{10}+46 x^{9}-103 x^{8}+113 x^{7}-38 x^{6}-77 x^{5}+133 x^{4}-101 x^{3}+43 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 158\)
\(\displaystyle a \! \left(7\right) = 478\)
\(\displaystyle a \! \left(8\right) = 1514\)
\(\displaystyle a \! \left(9\right) = 4998\)
\(\displaystyle a \! \left(10\right) = 17008\)
\(\displaystyle a \! \left(11\right) = 59109\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{6+n}+\frac{\left(7 n +10\right) a \! \left(1+n \right)}{6+n}-\frac{2 \left(3 n +8\right) a \! \left(n +2\right)}{6+n}-\frac{\left(24+7 n \right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(3 n +14\right) a \! \left(n +4\right)}{6+n}-\frac{n^{4}-12 n^{3}-11 n^{2}+126 n +80}{2 \left(6+n \right)}, \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 158\)
\(\displaystyle a \! \left(7\right) = 478\)
\(\displaystyle a \! \left(8\right) = 1514\)
\(\displaystyle a \! \left(9\right) = 4998\)
\(\displaystyle a \! \left(10\right) = 17008\)
\(\displaystyle a \! \left(11\right) = 59109\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{6+n}+\frac{\left(7 n +10\right) a \! \left(1+n \right)}{6+n}-\frac{2 \left(3 n +8\right) a \! \left(n +2\right)}{6+n}-\frac{\left(24+7 n \right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(3 n +14\right) a \! \left(n +4\right)}{6+n}-\frac{n^{4}-12 n^{3}-11 n^{2}+126 n +80}{2 \left(6+n \right)}, \quad n \geq 12\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 158 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
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Copy 158 equations to clipboard:
\(\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_{110}\! \left(x \right)+F_{4}\! \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_{108}\! \left(x \right)+F_{46}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{37}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{34}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x , 1\right)\\
F_{48}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{106}\! \left(x , y\right)+F_{49}\! \left(x , y\right)+F_{96}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{9}\! \left(x \right)\\
F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right) F_{57}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{54}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= y x\\
F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{53}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{13}\! \left(x \right)+F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{57}\! \left(x , y\right) F_{64}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{62}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{61}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{69}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{57}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{70}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{76}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{77}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{69}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{80}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{82}\! \left(x , y\right)+F_{84}\! \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_{62}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{57}\! \left(x , y\right) F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{23}\! \left(x \right)+F_{81}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= 3 F_{15}\! \left(x \right)+F_{87}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{88}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{89}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{91}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= 3 F_{15}\! \left(x \right)+F_{87}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{94}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{80}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{97}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= -\frac{-y F_{48}\! \left(x , y\right)+F_{48}\! \left(x , 1\right)}{-1+y}\\
F_{98}\! \left(x , y\right) &= F_{57}\! \left(x , y\right) F_{99}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{100}\! \left(x , y\right)+F_{104}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)+F_{103}\! \left(x , y\right)\\
F_{102}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{103}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{105}\! \left(x , y\right) &= -\frac{-y F_{99}\! \left(x , y\right)+F_{99}\! \left(x , 1\right)}{-1+y}\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{107}\! \left(x , y\right) &= -\frac{-y F_{99}\! \left(x , y\right)+F_{99}\! \left(x , 1\right)}{-1+y}\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{99}\! \left(x , 1\right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{114}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{12}\! \left(x \right) F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{128}\! \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_{127}\! \left(x \right) &= x^{2}\\
F_{128}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{12}\! \left(x \right) F_{122}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{152}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{146}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{12}\! \left(x \right) F_{135}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{12}\! \left(x \right) F_{139}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{12}\! \left(x \right) F_{144}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{143}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{12}\! \left(x \right) F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{15}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{153}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{12}\! \left(x \right) F_{155}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{157}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{154}\! \left(x \right)\\
\end{align*}\)