Av(1243, 1324, 1342, 2134, 4123)
Generating Function
\(\displaystyle \frac{\left(-3 x^{4}+x^{3}+3 x^{2}-3 x +1\right) \sqrt{1-4 x}-4 x^{5}+3 x^{4}+x^{3}-3 x^{2}+3 x -1}{2 x \left(x^{2}+x -1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 179, 566, 1840, 6132, 20862, 72175, 253123, 897721, 3213778, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(x^{2}+x -1\right) \left(4 x^{5}-3 x^{4}-x^{3}+3 x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+4 x^{9}+3 x^{8}-8 x^{7}-8 x^{6}+18 x^{5}-4 x^{4}-13 x^{3}+14 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 179\)
\(\displaystyle a \! \left(7\right) = 566\)
\(\displaystyle a \! \left(8\right) = 1840\)
\(\displaystyle a \! \left(9\right) = 6132\)
\(\displaystyle a \! \left(n +8\right) = -\frac{6 \left(2 n +1\right) a \! \left(n \right)}{n +9}+\frac{\left(4+7 n \right) a \! \left(1+n \right)}{n +9}+\frac{\left(57+35 n \right) a \! \left(n +2\right)}{n +9}-\frac{\left(41 n +104\right) a \! \left(n +3\right)}{n +9}-\frac{\left(8 n +37\right) a \! \left(n +4\right)}{n +9}+\frac{\left(213+40 n \right) a \! \left(n +5\right)}{n +9}-\frac{\left(29 n +191\right) a \! \left(n +6\right)}{n +9}+\frac{\left(70+9 n \right) a \! \left(n +7\right)}{n +9}+\frac{2}{n +9}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 179\)
\(\displaystyle a \! \left(7\right) = 566\)
\(\displaystyle a \! \left(8\right) = 1840\)
\(\displaystyle a \! \left(9\right) = 6132\)
\(\displaystyle a \! \left(n +8\right) = -\frac{6 \left(2 n +1\right) a \! \left(n \right)}{n +9}+\frac{\left(4+7 n \right) a \! \left(1+n \right)}{n +9}+\frac{\left(57+35 n \right) a \! \left(n +2\right)}{n +9}-\frac{\left(41 n +104\right) a \! \left(n +3\right)}{n +9}-\frac{\left(8 n +37\right) a \! \left(n +4\right)}{n +9}+\frac{\left(213+40 n \right) a \! \left(n +5\right)}{n +9}-\frac{\left(29 n +191\right) a \! \left(n +6\right)}{n +9}+\frac{\left(70+9 n \right) a \! \left(n +7\right)}{n +9}+\frac{2}{n +9}, \quad n \geq 10\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 110 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_{21}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{104}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{21}\! \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_{101}\! \left(x , y\right)+F_{103}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{99}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{38}\! \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_{28}\! \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_{21}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= x\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{21}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{21}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{29}\! \left(x \right) &= 0\\
F_{30}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{23}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{23}\! \left(x \right)+F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x \right)+F_{50}\! \left(x , y\right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{21}\! \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_{1}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{21}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{21}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{50}\! \left(x , y\right) &= F_{29}\! \left(x \right)+F_{51}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{52}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{57}\! \left(x \right)+F_{79}\! \left(x , y\right)\\
F_{57}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{58}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{21}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{21}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{21}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{21}\! \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_{21}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{21}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{79}\! \left(x , y\right) &= F_{29}\! \left(x \right)+F_{80}\! \left(x , y\right)+F_{85}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= 2 F_{29}\! \left(x \right)+F_{80}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{86}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{62}\! \left(x \right)+F_{84}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{88}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{89}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{91}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{92}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= F_{95}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{96}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{97}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{95}\! \left(x , y\right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right) F_{21}\! \left(x \right)\\
F_{102}\! \left(x , y\right) &= \frac{F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{103}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{57}\! \left(x \right)\\
\end{align*}\)