Av(1243, 1342, 2134, 2314, 4123)
Generating Function
\(\displaystyle \frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{7}-x^{6}-x^{5}+x^{4}+3 x^{2}-3 x +1\right)}{2 x \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 165, 508, 1632, 5421, 18456, 63990, 224976, 799771, 2869084, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x
\right)^{2}+\left(x^{7}-x^{6}-x^{5}+x^{4}+3 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x^{7}-x^{6}-x^{5}+x^{4}+3 x^{2}-3 x +1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 165\)
\(\displaystyle a \! \left(7\right) = 508\)
\(\displaystyle a \! \left(8\right) = 1632\)
\(\displaystyle a \! \left(9\right) = 5421\)
\(\displaystyle a \! \left(10\right) = 18456\)
\(\displaystyle a \! \left(11\right) = 63990\)
\(\displaystyle a \! \left(12\right) = 224976\)
\(\displaystyle a \! \left(13\right) = 799771\)
\(\displaystyle a \! \left(14\right) = 2869084\)
\(\displaystyle a \! \left(n +9\right) = -\frac{2 \left(-7+2 n \right) a \! \left(n \right)}{10+n}+\frac{3 \left(-10+3 n \right) a \! \left(1+n \right)}{10+n}-\frac{2 \left(n -10\right) a \! \left(n +2\right)}{10+n}-\frac{8 \left(1+n \right) a \! \left(n +3\right)}{10+n}+\frac{6 \left(1+n \right) a \! \left(n +4\right)}{10+n}-\frac{\left(80+13 n \right) a \! \left(n +5\right)}{10+n}+\frac{9 \left(20+3 n \right) a \! \left(n +6\right)}{10+n}-\frac{2 \left(84+11 n \right) a \! \left(n +7\right)}{10+n}+\frac{2 \left(35+4 n \right) a \! \left(n +8\right)}{10+n}+\frac{n +4}{10+n}, \quad n \geq 15\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 165\)
\(\displaystyle a \! \left(7\right) = 508\)
\(\displaystyle a \! \left(8\right) = 1632\)
\(\displaystyle a \! \left(9\right) = 5421\)
\(\displaystyle a \! \left(10\right) = 18456\)
\(\displaystyle a \! \left(11\right) = 63990\)
\(\displaystyle a \! \left(12\right) = 224976\)
\(\displaystyle a \! \left(13\right) = 799771\)
\(\displaystyle a \! \left(14\right) = 2869084\)
\(\displaystyle a \! \left(n +9\right) = -\frac{2 \left(-7+2 n \right) a \! \left(n \right)}{10+n}+\frac{3 \left(-10+3 n \right) a \! \left(1+n \right)}{10+n}-\frac{2 \left(n -10\right) a \! \left(n +2\right)}{10+n}-\frac{8 \left(1+n \right) a \! \left(n +3\right)}{10+n}+\frac{6 \left(1+n \right) a \! \left(n +4\right)}{10+n}-\frac{\left(80+13 n \right) a \! \left(n +5\right)}{10+n}+\frac{9 \left(20+3 n \right) a \! \left(n +6\right)}{10+n}-\frac{2 \left(84+11 n \right) a \! \left(n +7\right)}{10+n}+\frac{2 \left(35+4 n \right) a \! \left(n +8\right)}{10+n}+\frac{n +4}{10+n}, \quad n \geq 15\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 93 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_{13}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{13}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{13}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{13}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{13}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{13}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{13}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{13}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{17}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{13}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{13}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x , 1\right)\\
F_{55}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{56}\! \left(x , y\right)+F_{90}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{58}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)+F_{83}\! \left(x \right)\\
F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{62}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{61}\! \left(x , y\right) F_{64}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= y x\\
F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{9}\! \left(x \right)\\
F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{64}\! \left(x , y\right) F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{69}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)+F_{76}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{19}\! \left(x \right)+F_{73}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{64}\! \left(x , y\right) F_{71}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{77}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= 2 F_{19}\! \left(x \right)+F_{78}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{79}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= 2 F_{19}\! \left(x \right)+F_{78}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{64}\! \left(x , y\right) F_{82}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{80}\! \left(x , y\right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{13}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{90}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{91}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= \frac{F_{55}\! \left(x , y\right) y -F_{55}\! \left(x , 1\right)}{-1+y}\\
F_{92}\! \left(x , y\right) &= F_{55}\! \left(x , y\right) F_{64}\! \left(x , y\right)\\
\end{align*}\)