Av(1342, 2341, 3124)
Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{3} \sqrt{-4 x +1}+x^{3}-x^{2}+3 x -1}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 265, 926, 3216, 11152, 38741, 135126, 473872, 1672151, 5939232, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)+x^{6}-6 x^{5}+16 x^{4}-22 x^{3}+18 x^{2}-7 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{\left(22+17 n \right) a \! \left(1+n \right)}{n +5}-\frac{5 \left(4 n +11\right) a \! \left(n +2\right)}{n +5}+\frac{\left(31+8 n \right) a \! \left(n +3\right)}{n +5}+\frac{9}{n +5}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{\left(22+17 n \right) a \! \left(1+n \right)}{n +5}-\frac{5 \left(4 n +11\right) a \! \left(n +2\right)}{n +5}+\frac{\left(31+8 n \right) a \! \left(n +3\right)}{n +5}+\frac{9}{n +5}, \quad n \geq 5\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 112 rules.
Found on July 23, 2021.Finding the specification took 2 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_{3}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{108}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\
F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{105}\! \left(x , y\right)+F_{107}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= y x\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x , y\right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{32}\! \left(x , y\right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{3}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x \right)+F_{49}\! \left(x , y\right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 0\\
F_{37}\! \left(x \right) &= F_{3}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{3}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{3}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{3}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{54}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{36}\! \left(x \right)+F_{55}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{56}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{53}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{59}\! \left(x \right)+F_{64}\! \left(x , y\right)\\
F_{59}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{3}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{3}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{64}\! \left(x , y\right) &= 2 F_{36}\! \left(x \right)+F_{65}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{66}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{58}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{70}\! \left(x \right)+F_{82}\! \left(x , y\right)\\
F_{70}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{3}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{3}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{3}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{3}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{82}\! \left(x , y\right) &= 2 F_{36}\! \left(x \right)+F_{83}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{84}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{69}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{3}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{3}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{3}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{3}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{101}\! \left(x , y\right) &= 3 F_{36}\! \left(x \right)+F_{102}\! \left(x , y\right)+F_{104}\! \left(x , y\right)\\
F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{103}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{86}\! \left(x , y\right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{106}\! \left(x , y\right) &= \frac{F_{7}\! \left(x , y\right) y -F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{107}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{63}\! \left(x \right)\\
\end{align*}\)