Av(1342, 2314, 4123)
Generating Function
\(\displaystyle -\frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{6}-16 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1\right) \left(x^{2}-x +1\right)}{2 x \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 243, 785, 2511, 8073, 26312, 87257, 294603, 1011602, 3526519, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(3 x^{3}-5 x^{2}+4 x -1\right)^{2} \left(x -1\right)^{10} F \left(x
\right)^{2}-\left(x^{2}-x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x^{6}-16 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)+\left(x^{2}-x +1\right)^{2} \left(x^{6}-16 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 243\)
\(\displaystyle a \! \left(7\right) = 785\)
\(\displaystyle a \! \left(8\right) = 2511\)
\(\displaystyle a \! \left(9\right) = 8073\)
\(\displaystyle a \! \left(10\right) = 26312\)
\(\displaystyle a \! \left(11\right) = 87257\)
\(\displaystyle a \! \left(12\right) = 294603\)
\(\displaystyle a \! \left(13\right) = 1011602\)
\(\displaystyle a \! \left(14\right) = 3526519\)
\(\displaystyle a \! \left(15\right) = 12456315\)
\(\displaystyle a \! \left(16\right) = 44495535\)
\(\displaystyle a \! \left(n +13\right) = \frac{6 \left(2 n +1\right) a \! \left(n \right)}{n +14}-\frac{\left(239 n +484\right) a \! \left(n +1\right)}{n +14}+\frac{3 \left(433 n +1220\right) a \! \left(n +2\right)}{n +14}-\frac{2 \left(1933 n +7145\right) a \! \left(n +3\right)}{n +14}+\frac{11 \left(691 n +3206\right) a \! \left(n +4\right)}{n +14}-\frac{6 \left(1781 n +10021\right) a \! \left(n +5\right)}{n +14}+\frac{2 \left(5574 n +36991\right) a \! \left(n +6\right)}{n +14}-\frac{2 \left(4376 n +33505\right) a \! \left(n +7\right)}{n +14}+\frac{2 \left(2580 n +22397\right) a \! \left(n +8\right)}{n +14}-\frac{2 \left(1123 n +10904\right) a \! \left(n +9\right)}{n +14}+\frac{\left(7492+697 n \right) a \! \left(n +10\right)}{n +14}-\frac{\left(145 n +1712\right) a \! \left(n +11\right)}{n +14}+\frac{2 \left(9 n +116\right) a \! \left(n +12\right)}{n +14}+\frac{\left(n -4\right) \left(n^{2}+n +6\right)}{6 n +84}, \quad n \geq 17\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 243\)
\(\displaystyle a \! \left(7\right) = 785\)
\(\displaystyle a \! \left(8\right) = 2511\)
\(\displaystyle a \! \left(9\right) = 8073\)
\(\displaystyle a \! \left(10\right) = 26312\)
\(\displaystyle a \! \left(11\right) = 87257\)
\(\displaystyle a \! \left(12\right) = 294603\)
\(\displaystyle a \! \left(13\right) = 1011602\)
\(\displaystyle a \! \left(14\right) = 3526519\)
\(\displaystyle a \! \left(15\right) = 12456315\)
\(\displaystyle a \! \left(16\right) = 44495535\)
\(\displaystyle a \! \left(n +13\right) = \frac{6 \left(2 n +1\right) a \! \left(n \right)}{n +14}-\frac{\left(239 n +484\right) a \! \left(n +1\right)}{n +14}+\frac{3 \left(433 n +1220\right) a \! \left(n +2\right)}{n +14}-\frac{2 \left(1933 n +7145\right) a \! \left(n +3\right)}{n +14}+\frac{11 \left(691 n +3206\right) a \! \left(n +4\right)}{n +14}-\frac{6 \left(1781 n +10021\right) a \! \left(n +5\right)}{n +14}+\frac{2 \left(5574 n +36991\right) a \! \left(n +6\right)}{n +14}-\frac{2 \left(4376 n +33505\right) a \! \left(n +7\right)}{n +14}+\frac{2 \left(2580 n +22397\right) a \! \left(n +8\right)}{n +14}-\frac{2 \left(1123 n +10904\right) a \! \left(n +9\right)}{n +14}+\frac{\left(7492+697 n \right) a \! \left(n +10\right)}{n +14}-\frac{\left(145 n +1712\right) a \! \left(n +11\right)}{n +14}+\frac{2 \left(9 n +116\right) a \! \left(n +12\right)}{n +14}+\frac{\left(n -4\right) \left(n^{2}+n +6\right)}{6 n +84}, \quad n \geq 17\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 134 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_{22}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{126}\! \left(x \right)+F_{4}\! \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_{123}\! \left(x , y\right)+F_{125}\! \left(x , y\right)+F_{7}\! \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_{80}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{35}\! \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_{31}\! \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_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= 0\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{22}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{20}\! \left(x \right)+F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{22}\! \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_{1}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{22}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{22}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{22}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{22}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{55}\! \left(x , y\right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{22}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{22}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{22}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{22}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{22}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{22}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{22}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{80}\! \left(x , y\right) &= F_{117}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{20}\! \left(x \right)+F_{83}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{84}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{82}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{89}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{22}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{22}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{22}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{25}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{103}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{104}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{109}\! \left(x , y\right) &= 2 F_{25}\! \left(x \right)+F_{110}\! \left(x , y\right)+F_{112}\! \left(x , y\right)\\
F_{110}\! \left(x , y\right) &= F_{111}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{111}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)+F_{98}\! \left(x \right)\\
F_{114}\! \left(x , y\right) &= 2 F_{25}\! \left(x \right)+F_{112}\! \left(x , y\right)+F_{115}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{116}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{119}\! \left(x , y\right)\\
F_{118}\! \left(x , y\right) &= F_{42}\! \left(x \right)\\
F_{119}\! \left(x , y\right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{121}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{123}\! \left(x , y\right) &= F_{124}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{124}\! \left(x , y\right) &= \frac{F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{125}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{130}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{42}\! \left(x \right)\\
\end{align*}\)