Av(1324, 1342, 4123)
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Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{3} \sqrt{-4 x +1}+x^{3}-x^{2}+3 x -1}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 21, 75, 265, 926, 3216, 11152, 38741, 135126, 473872, 1672151, 5939232, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \left(x \right)^{2}-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)+x^{6}-6 x^{5}+16 x^{4}-22 x^{3}+18 x^{2}-7 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{\left(22+17 n \right) a \! \left(1+n \right)}{n +5}-\frac{5 \left(4 n +11\right) a \! \left(n +2\right)}{n +5}+\frac{\left(31+8 n \right) a \! \left(n +3\right)}{n +5}+\frac{9}{n +5}, \quad n \geq 5\)
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This specification was found using the strategy pack "Row And Col Placements Expand Verified" and has 45 rules.

Found on January 21, 2022.

Finding the specification took 72 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_{43}\! \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_{4}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{3}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{10}\! \left(x \right) &= \frac{F_{11}\! \left(x \right)}{F_{3}\! \left(x \right)}\\ F_{11}\! \left(x \right) &= -F_{15}\! \left(x \right)-F_{33}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{3}\! \left(x \right)}\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= -F_{30}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{18}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\ F_{22}\! \left(x , y\right) &= \frac{y F_{23}\! \left(x , y\right)-F_{23}\! \left(x , 1\right)}{-1+y}\\ F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= y x\\ F_{26}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{3}\! \left(x \right)}\\ F_{29}\! \left(x \right) &= -F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{3}\! \left(x \right) F_{30}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{3}\! \left(x \right)}\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= -F_{16}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{3}\! \left(x \right) F_{30}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{42}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{28}\! \left(x \right) F_{3}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{3}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{4}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row And Col Placements Expand Verified" and has 103 rules.

Found on January 21, 2022.

Finding the specification took 96 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_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{100}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5}\! \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_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{12}\! \left(x \right) &= -F_{18}\! \left(x \right)-F_{37}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= -F_{19}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{17}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{33}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{30}\! \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_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x , 1\right)\\ F_{25}\! \left(x , y\right) &= -\frac{-y F_{26}\! \left(x , y\right)+F_{26}\! \left(x , 1\right)}{-1+y}\\ F_{26}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= y x\\ F_{29}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{32}\! \left(x \right) &= -F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{4}\! \left(x \right) F_{60}\! \left(x \right)}\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= -F_{49}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= \frac{F_{47}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= -F_{19}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= -F_{98}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= \frac{F_{56}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{56}\! \left(x \right) &= -F_{57}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{67}\! \left(x \right) &= 0\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{85}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= 2 F_{67}\! \left(x \right)+F_{92}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right) F_{40}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{15}\! \left(x \right)\\ \end{align*}\)