Av(1243, 1342, 2134, 4123)
Generating Function
\(\displaystyle \frac{\left(-x^{8}+3 x^{7}-2 x^{6}-x^{5}+8 x^{3}-10 x^{2}+5 x -1\right) \sqrt{1-4 x}-2 x^{9}+7 x^{8}-7 x^{7}+5 x^{5}+2 x^{4}-10 x^{3}+10 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 201, 638, 2064, 6822, 23009, 78984, 275192, 970789, 3460426, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \left(x
\right)^{2}+\left(x^{8}-3 x^{7}+2 x^{6}+x^{5}-2 x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{17}-6 x^{16}+13 x^{15}-10 x^{14}-6 x^{13}+16 x^{12}-12 x^{11}+20 x^{10}-38 x^{9}+33 x^{8}-23 x^{7}+71 x^{6}-150 x^{5}+167 x^{4}-110 x^{3}+44 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 201\)
\(\displaystyle a \! \left(7\right) = 638\)
\(\displaystyle a \! \left(8\right) = 2064\)
\(\displaystyle a \! \left(9\right) = 6822\)
\(\displaystyle a \! \left(10\right) = 23009\)
\(\displaystyle a \! \left(11\right) = 78984\)
\(\displaystyle a \! \left(12\right) = 275192\)
\(\displaystyle a \! \left(13\right) = 970789\)
\(\displaystyle a \! \left(14\right) = 3460426\)
\(\displaystyle a \! \left(15\right) = 12443613\)
\(\displaystyle a \! \left(16\right) = 45083237\)
\(\displaystyle a \! \left(17\right) = 164394500\)
\(\displaystyle a \! \left(n +11\right) = \frac{4 \left(-7+2 n \right) a \! \left(n \right)}{n +12}-\frac{2 \left(-49+19 n \right) a \! \left(n +1\right)}{n +12}+\frac{\left(-126+65 n \right) a \! \left(n +2\right)}{n +12}-\frac{6 \left(-15+7 n \right) a \! \left(n +3\right)}{n +12}+\frac{3 \left(n -12\right) a \! \left(n +4\right)}{n +12}-\frac{\left(406+59 n \right) a \! \left(n +5\right)}{n +12}+\frac{\left(1302+191 n \right) a \! \left(n +6\right)}{n +12}-\frac{4 \left(446+59 n \right) a \! \left(n +7\right)}{n +12}+\frac{6 \left(223+26 n \right) a \! \left(n +8\right)}{n +12}-\frac{\left(572+59 n \right) a \! \left(n +9\right)}{n +12}+\frac{2 \left(65+6 n \right) a \! \left(n +10\right)}{n +12}+\frac{2 n +2}{n +12}, \quad n \geq 18\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 201\)
\(\displaystyle a \! \left(7\right) = 638\)
\(\displaystyle a \! \left(8\right) = 2064\)
\(\displaystyle a \! \left(9\right) = 6822\)
\(\displaystyle a \! \left(10\right) = 23009\)
\(\displaystyle a \! \left(11\right) = 78984\)
\(\displaystyle a \! \left(12\right) = 275192\)
\(\displaystyle a \! \left(13\right) = 970789\)
\(\displaystyle a \! \left(14\right) = 3460426\)
\(\displaystyle a \! \left(15\right) = 12443613\)
\(\displaystyle a \! \left(16\right) = 45083237\)
\(\displaystyle a \! \left(17\right) = 164394500\)
\(\displaystyle a \! \left(n +11\right) = \frac{4 \left(-7+2 n \right) a \! \left(n \right)}{n +12}-\frac{2 \left(-49+19 n \right) a \! \left(n +1\right)}{n +12}+\frac{\left(-126+65 n \right) a \! \left(n +2\right)}{n +12}-\frac{6 \left(-15+7 n \right) a \! \left(n +3\right)}{n +12}+\frac{3 \left(n -12\right) a \! \left(n +4\right)}{n +12}-\frac{\left(406+59 n \right) a \! \left(n +5\right)}{n +12}+\frac{\left(1302+191 n \right) a \! \left(n +6\right)}{n +12}-\frac{4 \left(446+59 n \right) a \! \left(n +7\right)}{n +12}+\frac{6 \left(223+26 n \right) a \! \left(n +8\right)}{n +12}-\frac{\left(572+59 n \right) a \! \left(n +9\right)}{n +12}+\frac{2 \left(65+6 n \right) a \! \left(n +10\right)}{n +12}+\frac{2 n +2}{n +12}, \quad n \geq 18\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 130 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_{21}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{116}\! \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_{113}\! \left(x , y\right)+F_{115}\! \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_{50}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{43}\! \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_{31}\! \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_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{27}\! \left(x \right) &= 0\\
F_{28}\! \left(x \right) &= F_{21}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{21}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{31}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{37}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{23}\! \left(x \right)\\
F_{40}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{23}\! \left(x \right)+F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{45}\! \left(x \right)+F_{58}\! \left(x , y\right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{21}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{21}\! \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_{54}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{21}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{20}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{25}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{58}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{59}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{60}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{44}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{65}\! \left(x \right)+F_{89}\! \left(x , y\right)\\
F_{65}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{66}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{21}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{21}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{21}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{21}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{83}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{21}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{21}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{89}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{90}\! \left(x , y\right)+F_{95}\! \left(x , y\right)+F_{97}\! \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_{42}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{94}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= 2 F_{27}\! \left(x \right)+F_{90}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{96}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{70}\! \left(x \right)+F_{94}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{98}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{105}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)\\
F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)\\
F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right) F_{21}\! \left(x \right)\\
F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{104}\! \left(x , y\right)\\
F_{103}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)\\
F_{106}\! \left(x , y\right) &= 2 F_{27}\! \left(x \right)+F_{107}\! \left(x , y\right)+F_{111}\! \left(x , y\right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{21}\! \left(x \right)\\
F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{110}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\
F_{110}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)\\
F_{111}\! \left(x , y\right) &= F_{112}\! \left(x , y\right) F_{21}\! \left(x \right)\\
F_{112}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{106}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right) F_{21}\! \left(x \right)\\
F_{114}\! \left(x , y\right) &= \frac{F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{115}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{20}\! \left(x \right) F_{21}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{55}\! \left(x \right)\\
\end{align*}\)