Av(1243, 2314, 4123)
Generating Function
\(\displaystyle \frac{\left(-2 x^{6}+x^{5}+7 x^{4}-13 x^{3}+11 x^{2}-5 x +1\right) \sqrt{1-4 x}+6 x^{6}-3 x^{5}-5 x^{4}+9 x^{3}-9 x^{2}+5 x -1}{2 x \left(2 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 21, 74, 254, 858, 2889, 9775, 33371, 115135, 401538, 1414821, 5032193, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{8} F \left(x
\right)^{2}-\left(2 x -1\right) \left(6 x^{6}-3 x^{5}-5 x^{4}+9 x^{3}-9 x^{2}+5 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{12}+4 x^{11}-35 x^{10}+60 x^{9}-3 x^{8}-169 x^{7}+350 x^{6}-399 x^{5}+302 x^{4}-157 x^{3}+54 x^{2}-11 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(6\right) = 254\)
\(\displaystyle a \! \left(7\right) = 858\)
\(\displaystyle a \! \left(8\right) = 2889\)
\(\displaystyle a \! \left(9\right) = 9775\)
\(\displaystyle a \! \left(10\right) = 33371\)
\(\displaystyle a \! \left(11\right) = 115135\)
\(\displaystyle a \! \left(12\right) = 401538\)
\(\displaystyle a \! \left(n +8\right) = \frac{8 \left(2 n -1\right) a \! \left(n \right)}{n +9}-\frac{4 \left(5 n -13\right) a \! \left(n +1\right)}{n +9}-\frac{6 \left(8 n +33\right) a \! \left(n +2\right)}{n +9}+\frac{\left(529+145 n \right) a \! \left(n +3\right)}{n +9}-\frac{\left(173 n +791\right) a \! \left(n +4\right)}{n +9}+\frac{7 \left(17 n +97\right) a \! \left(n +5\right)}{n +9}-\frac{\left(49 n +335\right) a \! \left(n +6\right)}{n +9}+\frac{\left(87+11 n \right) a \! \left(n +7\right)}{n +9}+\frac{n^{3}-12 n^{2}+47 n -48}{2 n +18}, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(6\right) = 254\)
\(\displaystyle a \! \left(7\right) = 858\)
\(\displaystyle a \! \left(8\right) = 2889\)
\(\displaystyle a \! \left(9\right) = 9775\)
\(\displaystyle a \! \left(10\right) = 33371\)
\(\displaystyle a \! \left(11\right) = 115135\)
\(\displaystyle a \! \left(12\right) = 401538\)
\(\displaystyle a \! \left(n +8\right) = \frac{8 \left(2 n -1\right) a \! \left(n \right)}{n +9}-\frac{4 \left(5 n -13\right) a \! \left(n +1\right)}{n +9}-\frac{6 \left(8 n +33\right) a \! \left(n +2\right)}{n +9}+\frac{\left(529+145 n \right) a \! \left(n +3\right)}{n +9}-\frac{\left(173 n +791\right) a \! \left(n +4\right)}{n +9}+\frac{7 \left(17 n +97\right) a \! \left(n +5\right)}{n +9}-\frac{\left(49 n +335\right) a \! \left(n +6\right)}{n +9}+\frac{\left(87+11 n \right) a \! \left(n +7\right)}{n +9}+\frac{n^{3}-12 n^{2}+47 n -48}{2 n +18}, \quad n \geq 13\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 153 rules.
Found on July 23, 2021.Finding the specification took 3 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{106}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \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_{22}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{12}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{12}\! \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_{20}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{12}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{12}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{12}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{12}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{12}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{87}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{12}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{12}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{12}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{102}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{12}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{101}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x , 1\right)\\
F_{108}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{109}\! \left(x , y\right)+F_{150}\! \left(x , y\right)+F_{152}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{110}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)+F_{127}\! \left(x , y\right)\\
F_{111}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)+F_{120}\! \left(x , y\right)\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)+F_{117}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{114}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= F_{113}\! \left(x , y\right) F_{116}\! \left(x , y\right)\\
F_{116}\! \left(x , y\right) &= y x\\
F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{8}\! \left(x \right)\\
F_{118}\! \left(x , y\right) &= F_{119}\! \left(x , y\right)\\
F_{119}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{117}\! \left(x , y\right)\\
F_{120}\! \left(x , y\right) &= F_{121}\! \left(x , y\right)+F_{124}\! \left(x , y\right)\\
F_{121}\! \left(x , y\right) &= F_{122}\! \left(x , y\right)+F_{22}\! \left(x \right)\\
F_{122}\! \left(x , y\right) &= F_{123}\! \left(x , y\right)\\
F_{123}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{121}\! \left(x , y\right)\\
F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)+F_{40}\! \left(x \right)\\
F_{125}\! \left(x , y\right) &= F_{126}\! \left(x , y\right)\\
F_{126}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{124}\! \left(x , y\right)\\
F_{127}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)+F_{139}\! \left(x , y\right)\\
F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right)+F_{134}\! \left(x , y\right)\\
F_{129}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)+F_{29}\! \left(x \right)\\
F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)+F_{133}\! \left(x , y\right)+F_{15}\! \left(x \right)\\
F_{131}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{132}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)+F_{130}\! \left(x , y\right)\\
F_{133}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{129}\! \left(x , y\right)\\
F_{134}\! \left(x , y\right) &= F_{135}\! \left(x , y\right)+F_{55}\! \left(x \right)\\
F_{135}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{136}\! \left(x , y\right)+F_{138}\! \left(x , y\right)\\
F_{136}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{137}\! \left(x , y\right)\\
F_{137}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{135}\! \left(x , y\right)\\
F_{138}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{134}\! \left(x , y\right)\\
F_{139}\! \left(x , y\right) &= F_{140}\! \left(x , y\right)+F_{145}\! \left(x , y\right)\\
F_{140}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)+F_{75}\! \left(x \right)\\
F_{141}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{142}\! \left(x , y\right)+F_{144}\! \left(x , y\right)\\
F_{142}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{143}\! \left(x , y\right)\\
F_{143}\! \left(x , y\right) &= F_{122}\! \left(x , y\right)+F_{141}\! \left(x , y\right)\\
F_{144}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{140}\! \left(x , y\right)\\
F_{145}\! \left(x , y\right) &= F_{146}\! \left(x , y\right)+F_{87}\! \left(x \right)\\
F_{146}\! \left(x , y\right) &= 3 F_{15}\! \left(x \right)+F_{147}\! \left(x , y\right)+F_{149}\! \left(x , y\right)\\
F_{147}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{148}\! \left(x , y\right)\\
F_{148}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)+F_{146}\! \left(x , y\right)\\
F_{149}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{145}\! \left(x , y\right)\\
F_{150}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{151}\! \left(x , y\right)\\
F_{151}\! \left(x , y\right) &= \frac{F_{108}\! \left(x , y\right) y -F_{108}\! \left(x , 1\right)}{-1+y}\\
F_{152}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{116}\! \left(x , y\right)\\
\end{align*}\)