Av(1342, 2341, 4123)
Generating Function
\(\displaystyle \frac{-2 \left(x -\frac{1}{2}\right) \left(x -1\right)^{4} \sqrt{1-4 x}-8 x^{5}+3 x^{4}+8 x^{3}-12 x^{2}+6 x -1}{12 x^{6}-38 x^{5}+52 x^{4}-38 x^{3}+14 x^{2}-2 x}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 262, 895, 3022, 10188, 34524, 118030, 407754, 1423886, 5023900, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{2} \left(2 x -1\right)^{2} \left(3 x^{3}-5 x^{2}+4 x -1\right)^{2} F \left(x
\right)^{2}+\left(x -1\right) \left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(8 x^{5}-3 x^{4}-8 x^{3}+12 x^{2}-6 x +1\right) F \! \left(x \right)+4 x^{10}-21 x^{9}+142 x^{8}-410 x^{7}+678 x^{6}-719 x^{5}+511 x^{4}-243 x^{3}+74 x^{2}-13 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 262\)
\(\displaystyle a \! \left(7\right) = 895\)
\(\displaystyle a \! \left(n +7\right) = \frac{24 \left(1+2 n \right) a \! \left(n \right)}{8+n}-\frac{4 \left(88+47 n \right) a \! \left(1+n \right)}{8+n}+\frac{2 \left(477+164 n \right) a \! \left(n +2\right)}{8+n}-\frac{\left(1268+327 n \right) a \! \left(n +3\right)}{8+n}+\frac{2 \left(477+98 n \right) a \! \left(n +4\right)}{8+n}-\frac{\left(406+69 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(90+13 n \right) a \! \left(n +6\right)}{8+n}+\frac{24}{8+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 262\)
\(\displaystyle a \! \left(7\right) = 895\)
\(\displaystyle a \! \left(n +7\right) = \frac{24 \left(1+2 n \right) a \! \left(n \right)}{8+n}-\frac{4 \left(88+47 n \right) a \! \left(1+n \right)}{8+n}+\frac{2 \left(477+164 n \right) a \! \left(n +2\right)}{8+n}-\frac{\left(1268+327 n \right) a \! \left(n +3\right)}{8+n}+\frac{2 \left(477+98 n \right) a \! \left(n +4\right)}{8+n}-\frac{\left(406+69 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(90+13 n \right) a \! \left(n +6\right)}{8+n}+\frac{24}{8+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 95 rules.
Found on July 23, 2021.Finding the specification took 5 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_{5}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= y x\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)+F_{85}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{69}\! \left(x , y\right)+F_{71}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x , y\right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{15}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x , y\right)\\
F_{25}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{25}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{16}\! \left(x \right)+F_{29}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x \right)+F_{40}\! \left(x , y\right)\\
F_{36}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\
F_{42}\! \left(x \right) &= 0\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{44}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{16}\! \left(x \right)+F_{41}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{49}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{51}\! \left(x \right)+F_{60}\! \left(x , y\right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{5}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{5}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{5}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= 2 F_{42}\! \left(x \right)+F_{62}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{63}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{65}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{52}\! \left(x \right)+F_{61}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{70}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= -\frac{-y F_{10}\! \left(x , y\right)+F_{10}\! \left(x , 1\right)}{-1+y}\\
F_{71}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{73}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)+F_{76}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{78}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= -\frac{-y F_{72}\! \left(x , y\right)+F_{72}\! \left(x , 1\right)}{-1+y}\\
F_{79}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{80}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{56}\! \left(x \right)+F_{82}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{82}\! \left(x , y\right)\\
F_{85}\! \left(x \right) &= F_{5}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{85}\! \left(x \right)+F_{1}\! \left(x \right)+F_{8}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{15}\! \left(x \right) F_{5}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{90}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x , 1\right)\\
F_{91}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{72}\! \left(x , y\right)\\
F_{92}\! \left(x \right) &= F_{5}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{5}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{72}\! \left(x , 1\right)\\
\end{align*}\)