Av(1324, 1342, 1423, 2341)
Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \sqrt{-4 x +1}-2 x^{2}-2 x +1}{2 \left(x^{2}-3 x +1\right) x}\)
Counting Sequence
1, 1, 2, 6, 20, 68, 232, 793, 2719, 9366, 32451, 113181, 397532, 1406291, 5009981, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}-3 x +1\right)^{2} F \left(x
\right)^{2}+\left(x^{2}-3 x +1\right) \left(2 x^{2}+2 x -1\right) F \! \left(x \right)+x^{3}+6 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(3+2 n \right) a \! \left(n \right)}{5+n}+\frac{2 \left(28+15 n \right) a \! \left(1+n \right)}{5+n}-\frac{\left(77+27 n \right) a \! \left(n +2\right)}{5+n}+\frac{\left(35+9 n \right) a \! \left(n +3\right)}{5+n}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(3+2 n \right) a \! \left(n \right)}{5+n}+\frac{2 \left(28+15 n \right) a \! \left(1+n \right)}{5+n}-\frac{\left(77+27 n \right) a \! \left(n +2\right)}{5+n}+\frac{\left(35+9 n \right) a \! \left(n +3\right)}{5+n}, \quad n \geq 4\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 174 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{173}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{7}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= y x\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{11}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= \frac{F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x \right)+F_{89}\! \left(x , y\right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \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_{11}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{11}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{34}\! \left(x \right) &= 0\\
F_{35}\! \left(x \right) &= F_{11}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{11}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{41}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{11}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{34}\! \left(x \right)+F_{48}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{11}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{11}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{11}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{53}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{11}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{60}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{11}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{11}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 2 F_{34}\! \left(x \right)+F_{67}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{11}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{11}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{11}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 2 F_{34}\! \left(x \right)+F_{75}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{11}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= 3 F_{34}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{11}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{11}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{11}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{89}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{91}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{92}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{93}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{94}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{92}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{34}\! \left(x \right)+F_{96}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{97}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{95}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{91}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{103}\! \left(x , y\right)\\
F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)\\
F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right) F_{11}\! \left(x \right)\\
F_{102}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{11}\! \left(x \right)\\
F_{105}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)+F_{142}\! \left(x , y\right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{121}\! \left(x , y\right)\\
F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{110}\! \left(x , y\right)\\
F_{110}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)+F_{114}\! \left(x , y\right)\\
F_{111}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{92}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{115}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= 2 F_{34}\! \left(x \right)+F_{116}\! \left(x , y\right)+F_{120}\! \left(x , y\right)\\
F_{116}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{117}\! \left(x , y\right)\\
F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{119}\! \left(x , y\right)\\
F_{118}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)\\
F_{119}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)\\
F_{120}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{11}\! \left(x \right)\\
F_{121}\! \left(x , y\right) &= 2 F_{34}\! \left(x \right)+F_{122}\! \left(x , y\right)+F_{134}\! \left(x , y\right)\\
F_{122}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{123}\! \left(x , y\right)\\
F_{123}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)+F_{127}\! \left(x , y\right)\\
F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\
F_{125}\! \left(x , y\right) &= F_{126}\! \left(x , y\right)\\
F_{126}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{95}\! \left(x , y\right)\\
F_{127}\! \left(x , y\right) &= F_{121}\! \left(x , y\right)+F_{128}\! \left(x , y\right)\\
F_{128}\! \left(x , y\right) &= 3 F_{34}\! \left(x \right)+F_{129}\! \left(x , y\right)+F_{133}\! \left(x , y\right)\\
F_{129}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{130}\! \left(x , y\right)\\
F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)+F_{132}\! \left(x , y\right)\\
F_{131}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)\\
F_{133}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{121}\! \left(x , y\right)\\
F_{134}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{135}\! \left(x , y\right)\\
F_{135}\! \left(x , y\right) &= F_{136}\! \left(x , y\right)+F_{139}\! \left(x , y\right)\\
F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)\\
F_{137}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{138}\! \left(x , y\right)\\
F_{138}\! \left(x , y\right) &= F_{136}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{139}\! \left(x , y\right) &= 2 F_{34}\! \left(x \right)+F_{134}\! \left(x , y\right)+F_{140}\! \left(x , y\right)\\
F_{140}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{141}\! \left(x , y\right)\\
F_{141}\! \left(x , y\right) &= F_{139}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\
F_{142}\! \left(x , y\right) &= F_{143}\! \left(x , y\right)+F_{158}\! \left(x , y\right)\\
F_{143}\! \left(x , y\right) &= 2 F_{34}\! \left(x \right)+F_{144}\! \left(x , y\right)+F_{156}\! \left(x , y\right)\\
F_{144}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{145}\! \left(x , y\right)\\
F_{145}\! \left(x , y\right) &= F_{146}\! \left(x , y\right)+F_{149}\! \left(x , y\right)\\
F_{146}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{147}\! \left(x , y\right)\\
F_{147}\! \left(x , y\right) &= F_{148}\! \left(x , y\right)\\
F_{148}\! \left(x , y\right) &= F_{100}\! \left(x , y\right) F_{11}\! \left(x \right)\\
F_{149}\! \left(x , y\right) &= F_{143}\! \left(x , y\right)+F_{150}\! \left(x , y\right)\\
F_{150}\! \left(x , y\right) &= 3 F_{34}\! \left(x \right)+F_{151}\! \left(x , y\right)+F_{155}\! \left(x , y\right)\\
F_{151}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{152}\! \left(x , y\right)\\
F_{152}\! \left(x , y\right) &= F_{153}\! \left(x , y\right)+F_{154}\! \left(x , y\right)\\
F_{153}\! \left(x , y\right) &= F_{147}\! \left(x , y\right)\\
F_{154}\! \left(x , y\right) &= F_{150}\! \left(x , y\right)\\
F_{155}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{143}\! \left(x , y\right)\\
F_{156}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{157}\! \left(x , y\right)\\
F_{157}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{143}\! \left(x , y\right)\\
F_{158}\! \left(x , y\right) &= 3 F_{34}\! \left(x \right)+F_{159}\! \left(x , y\right)+F_{171}\! \left(x , y\right)\\
F_{159}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{160}\! \left(x , y\right)\\
F_{160}\! \left(x , y\right) &= F_{161}\! \left(x , y\right)+F_{164}\! \left(x , y\right)\\
F_{161}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{162}\! \left(x , y\right)\\
F_{162}\! \left(x , y\right) &= F_{163}\! \left(x , y\right)\\
F_{163}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{11}\! \left(x \right)\\
F_{164}\! \left(x , y\right) &= F_{158}\! \left(x , y\right)+F_{165}\! \left(x , y\right)\\
F_{165}\! \left(x , y\right) &= 4 F_{34}\! \left(x \right)+F_{166}\! \left(x , y\right)+F_{170}\! \left(x , y\right)\\
F_{166}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{167}\! \left(x , y\right)\\
F_{167}\! \left(x , y\right) &= F_{168}\! \left(x , y\right)+F_{169}\! \left(x , y\right)\\
F_{168}\! \left(x , y\right) &= F_{162}\! \left(x , y\right)\\
F_{169}\! \left(x , y\right) &= F_{165}\! \left(x , y\right)\\
F_{170}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{158}\! \left(x , y\right)\\
F_{171}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{172}\! \left(x , y\right)\\
F_{172}\! \left(x , y\right) &= F_{121}\! \left(x , y\right)+F_{158}\! \left(x , y\right)\\
F_{173}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
\end{align*}\)