Av(1324, 1342, 2314, 4123)
Generating Function
\(\displaystyle -\frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{5}-5 x^{4}+9 x^{3}-10 x^{2}+5 x -1\right)}{2 x \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 197, 603, 1883, 6065, 20147, 68666, 238703, 842299, 3006559, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{10} F \left(x
\right)^{2}-\left(x^{5}-5 x^{4}+9 x^{3}-10 x^{2}+5 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+\left(x^{5}-5 x^{4}+9 x^{3}-10 x^{2}+5 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 197\)
\(\displaystyle a \! \left(7\right) = 603\)
\(\displaystyle a \! \left(8\right) = 1883\)
\(\displaystyle a \! \left(9\right) = 6065\)
\(\displaystyle a \! \left(10\right) = 20147\)
\(\displaystyle a \! \left(n +7\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +8}-\frac{\left(25 n +38\right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(31 n +75\right) a \! \left(n +2\right)}{n +8}-\frac{2 \left(45 n +166\right) a \! \left(n +3\right)}{n +8}+\frac{\left(368+79 n \right) a \! \left(n +4\right)}{n +8}-\frac{3 \left(13 n +74\right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(5 n +34\right) a \! \left(n +6\right)}{n +8}+\frac{\left(n +4\right) \left(n +3\right) \left(n +2\right)}{6 n +48}, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 197\)
\(\displaystyle a \! \left(7\right) = 603\)
\(\displaystyle a \! \left(8\right) = 1883\)
\(\displaystyle a \! \left(9\right) = 6065\)
\(\displaystyle a \! \left(10\right) = 20147\)
\(\displaystyle a \! \left(n +7\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +8}-\frac{\left(25 n +38\right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(31 n +75\right) a \! \left(n +2\right)}{n +8}-\frac{2 \left(45 n +166\right) a \! \left(n +3\right)}{n +8}+\frac{\left(368+79 n \right) a \! \left(n +4\right)}{n +8}-\frac{3 \left(13 n +74\right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(5 n +34\right) a \! \left(n +6\right)}{n +8}+\frac{\left(n +4\right) \left(n +3\right) \left(n +2\right)}{6 n +48}, \quad n \geq 11\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 107 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_{22}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{22}\! \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_{7}\! \left(x , y\right)+F_{96}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= y x\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x \right)+F_{26}\! \left(x , y\right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= x\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{22}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{22}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{22}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{74}\! \left(x , y\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_{20}\! \left(x \right)+F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\
F_{43}\! \left(x \right) &= 0\\
F_{44}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x \right)+F_{66}\! \left(x , y\right)\\
F_{48}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{22}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{22}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{22}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{22}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{22}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{22}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{66}\! \left(x , y\right) &= 2 F_{43}\! \left(x \right)+F_{67}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{70}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{58}\! \left(x \right)+F_{71}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= 2 F_{43}\! \left(x \right)+F_{69}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{73}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{22}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{22}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{22}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{84}\! \left(x , y\right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= 2 F_{43}\! \left(x \right)+F_{86}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{22}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{22}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{22}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{22}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{96}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{97}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= \frac{F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{98}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{85}\! \left(x \right)\\
\end{align*}\)