Av(1324, 1342, 2341, 4123)
Generating Function
\(\displaystyle \frac{-2 \left(x -\frac{1}{2}\right) \left(-1+x \right)^{5} \sqrt{1-4 x}-2 x^{7}+2 x^{6}-13 x^{5}+27 x^{4}-30 x^{3}+20 x^{2}-7 x +1}{2 x \left(2 x -1\right) \left(-1+x \right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 195, 585, 1783, 5617, 18356, 61945, 214340, 755512, 2699533, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(-1+x \right)^{10} F \left(x
\right)^{2}+\left(2 x -1\right) \left(2 x^{7}-2 x^{6}+13 x^{5}-27 x^{4}+30 x^{3}-20 x^{2}+7 x -1\right) \left(-1+x \right)^{5} F \! \left(x \right)+x^{13}+2 x^{12}-31 x^{11}+192 x^{10}-626 x^{9}+1307 x^{8}-1905 x^{7}+2007 x^{6}-1545 x^{5}+862 x^{4}-339 x^{3}+89 x^{2}-14 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 195\)
\(\displaystyle a \! \left(7\right) = 585\)
\(\displaystyle a \! \left(8\right) = 1783\)
\(\displaystyle a \! \left(9\right) = 5617\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +4}-\frac{4 \left(5 n +8\right) a \! \left(n +1\right)}{n +4}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{n +4}-\frac{n \left(n -1\right) \left(n -6\right) \left(3 n^{2}-43 n +2\right)}{24 \left(n +4\right)}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 195\)
\(\displaystyle a \! \left(7\right) = 585\)
\(\displaystyle a \! \left(8\right) = 1783\)
\(\displaystyle a \! \left(9\right) = 5617\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +4}-\frac{4 \left(5 n +8\right) a \! \left(n +1\right)}{n +4}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{n +4}-\frac{n \left(n -1\right) \left(n -6\right) \left(3 n^{2}-43 n +2\right)}{24 \left(n +4\right)}, \quad n \geq 10\)
This specification was found using the strategy pack "Requirement Placements Tracked Fusion" and has 151 rules.
Found on July 23, 2021.Finding the specification took 15 seconds.
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\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{1}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{3}\! \left(x \right)\\
F_{2}\! \left(x \right) &= 1\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= x\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{23}\! \left(x \right) &= 0\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{41}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{5}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{53}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{5}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{5}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{5}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{3}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{64}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{5}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{69}\! \left(x \right)+F_{73}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{5}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{5}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{5}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{73}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{5}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{5}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{3}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{89}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{85}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{5}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{5}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{5}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{94}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{5}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{124}\! \left(x \right)+F_{23}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{5}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{106}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{107}\! \left(x \right)+F_{111}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= 3 F_{23}\! \left(x \right)+F_{111}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{123}\! \left(x \right) &= 0\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x , 1\right)\\
F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right)\\
F_{129}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)\\
F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right) F_{140}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{131}\! \left(x , y\right) &= F_{132}\! \left(x , y\right)+F_{150}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)+F_{95}\! \left(x \right)\\
F_{133}\! \left(x , y\right) &= F_{134}\! \left(x , y\right)+F_{141}\! \left(x , y\right)\\
F_{134}\! \left(x , y\right) &= F_{135}\! \left(x , y\right)\\
F_{135}\! \left(x , y\right) &= F_{136}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{136}\! \left(x , y\right) &= F_{134}\! \left(x , y\right)+F_{137}\! \left(x , y\right)\\
F_{137}\! \left(x , y\right) &= F_{138}\! \left(x , y\right)\\
F_{138}\! \left(x , y\right) &= F_{139}\! \left(x , y\right) F_{140}\! \left(x , y\right)\\
F_{139}\! \left(x , y\right) &= y x\\
F_{140}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{2}\! \left(x \right)\\
F_{141}\! \left(x , y\right) &= 2 F_{23}\! \left(x \right)+F_{142}\! \left(x , y\right)+F_{149}\! \left(x , y\right)\\
F_{142}\! \left(x , y\right) &= F_{143}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{143}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)+F_{144}\! \left(x , y\right)\\
F_{144}\! \left(x , y\right) &= F_{145}\! \left(x , y\right)+F_{147}\! \left(x , y\right)+F_{23}\! \left(x \right)\\
F_{145}\! \left(x , y\right) &= F_{139}\! \left(x , y\right) F_{146}\! \left(x , y\right)\\
F_{146}\! \left(x , y\right) &= F_{144}\! \left(x , y\right)+F_{3}\! \left(x \right)\\
F_{147}\! \left(x , y\right) &= F_{148}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{148}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{144}\! \left(x , y\right)\\
F_{149}\! \left(x , y\right) &= F_{133}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{150}\! \left(x , y\right) &= \frac{F_{129}\! \left(x , y\right) y -F_{129}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)