Av(1243, 1342, 4123)
Generating Function
\(\displaystyle \frac{\left(2 x^{4}-6 x^{3}+9 x^{2}-5 x +1\right) \sqrt{1-4 x}-6 x^{5}+24 x^{4}-36 x^{3}+27 x^{2}-9 x +1}{4 \left(x^{3}-4 x^{2}+5 x -1\right) \left(x -1\right)^{2} \left(x -\frac{1}{2}\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 76, 275, 993, 3593, 13068, 47838, 176277, 653538, 2436158, 9124352, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-4 x^{2}+5 x -1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(2 x -1\right) \left(6 x^{5}-24 x^{4}+36 x^{3}-27 x^{2}+9 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x \left(9 x^{6}-32 x^{5}+54 x^{4}-50 x^{3}+27 x^{2}-8 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 275\)
\(\displaystyle a \! \left(7\right) = 993\)
\(\displaystyle a \! \left(8\right) = 3593\)
\(\displaystyle a \! \left(9\right) = 13068\)
\(\displaystyle a \! \left(10\right) = 47838\)
\(\displaystyle a \! \left(n +10\right) = -\frac{8 \left(2 n +3\right) a \! \left(n \right)}{n +10}+\frac{4 \left(35 n +74\right) a \! \left(n +1\right)}{n +10}-\frac{2 \left(277 n +799\right) a \! \left(n +2\right)}{n +10}+\frac{2 \left(643 n +2413\right) a \! \left(n +3\right)}{n +10}-\frac{\left(1913 n +8883\right) a \! \left(n +4\right)}{n +10}+\frac{\left(10297+1866 n \right) a \! \left(n +5\right)}{n +10}-\frac{\left(1189 n +7608\right) a \! \left(n +6\right)}{n +10}+\frac{2 \left(243 n +1774\right) a \! \left(n +7\right)}{n +10}-\frac{2 \left(61 n +501\right) a \! \left(n +8\right)}{n +10}+\frac{\left(155+17 n \right) a \! \left(n +9\right)}{n +10}+\frac{1}{n +10}, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 275\)
\(\displaystyle a \! \left(7\right) = 993\)
\(\displaystyle a \! \left(8\right) = 3593\)
\(\displaystyle a \! \left(9\right) = 13068\)
\(\displaystyle a \! \left(10\right) = 47838\)
\(\displaystyle a \! \left(n +10\right) = -\frac{8 \left(2 n +3\right) a \! \left(n \right)}{n +10}+\frac{4 \left(35 n +74\right) a \! \left(n +1\right)}{n +10}-\frac{2 \left(277 n +799\right) a \! \left(n +2\right)}{n +10}+\frac{2 \left(643 n +2413\right) a \! \left(n +3\right)}{n +10}-\frac{\left(1913 n +8883\right) a \! \left(n +4\right)}{n +10}+\frac{\left(10297+1866 n \right) a \! \left(n +5\right)}{n +10}-\frac{\left(1189 n +7608\right) a \! \left(n +6\right)}{n +10}+\frac{2 \left(243 n +1774\right) a \! \left(n +7\right)}{n +10}-\frac{2 \left(61 n +501\right) a \! \left(n +8\right)}{n +10}+\frac{\left(155+17 n \right) a \! \left(n +9\right)}{n +10}+\frac{1}{n +10}, \quad n \geq 11\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 110 rules.
Found on July 23, 2021.Finding the specification took 8 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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{100}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\
F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{7}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= y x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{11}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{97}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\
F_{17}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{8}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{50}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{27}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{3}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{3}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 0\\
F_{41}\! \left(x \right) &= F_{3}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{3}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{3}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{27} \left(x \right)^{2} F_{3}\! \left(x \right)\\
F_{50}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{51}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{52}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{55}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{28}\! \left(x \right)+F_{58}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{40}\! \left(x \right)+F_{63}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{28}\! \left(x \right)+F_{62}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{66}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{31}\! \left(x \right)+F_{67}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{77}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= 2 F_{40}\! \left(x \right)+F_{80}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{83}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{86}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{88}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{27} \left(x \right)^{2} F_{3}\! \left(x \right) F_{57}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{90}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x , y\right)\\
F_{91}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{92}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= F_{95}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{96}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{98}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{91}\! \left(x \right)+F_{99}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{102}\! \left(x \right)+F_{15}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x , 1\right)\\
F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{28}\! \left(x \right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)+F_{109}\! \left(x , y\right)+F_{40}\! \left(x \right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{108}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
\end{align*}\)