Av(2413)
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Generating Function
\(\displaystyle \frac{-8 \sqrt{-8 x +1}\, x -8 x^{2}+\sqrt{-8 x +1}+20 x +1}{2 \left(x +1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 23, 103, 512, 2740, 15485, 91245, 555662, 3475090, 22214707, 144640291, 956560748, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x +1\right)^{3} F \left(x \right)^{2}+\left(8 x^{2}-20 x -1\right) F \! \left(x \right)+16 x = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(n +2\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{n +2}+\frac{\left(-8+7 n \right) a \! \left(n +1\right)}{n +2}, \quad n \geq 2\)
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This specification was found using the strategy pack "Requirement Placements Tracked Fusion" and has 91 rules.

Found on February 06, 2022.

Finding the specification took 14299 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_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= y x\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{18}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\ F_{17}\! \left(x \right) &= x\\ F_{19}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{18}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{18}\! \left(x , y\right) F_{24}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{24}\! \left(x , y\right) F_{78}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x \right) F_{6}\! \left(x , y\right)\\ F_{30}\! \left(x \right) &= \frac{F_{31}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{17}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{0}\! \left(x \right) F_{17}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= \frac{F_{40}\! \left(x \right)}{F_{17}\! \left(x \right) F_{53}\! \left(x \right)}\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{48}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= -F_{45}\! \left(x \right)+F_{0}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{17}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{17}\! \left(x \right) F_{45}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{57}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{60}\! \left(x , y\right) F_{78}\! \left(x \right)\\ F_{61}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{17}\! \left(x \right) F_{60}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{76}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{65}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{72}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= -\frac{-y F_{73}\! \left(x , y\right)+F_{73}\! \left(x , 1\right)}{-1+y}\\ F_{73}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{38}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{75}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right)\\ F_{77}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{6}\! \left(x , y\right)\\ F_{78}\! \left(x \right) &= \frac{F_{79}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{17}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{0}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{0}\! \left(x \right) F_{17}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{85}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{78}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{17}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{17}\! \left(x \right) F_{38}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{63}\! \left(x , 1\right)\\ \end{align*}\)