Av(1243, 1342, 1423, 2314, 3214)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x -1\right)^{2}}{2 x^{7}-4 x^{5}-x^{4}+6 x^{3}-8 x^{2}+5 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 54, 148, 410, 1149, 3227, 9051, 25369, 71118, 199417, 559216, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{7}-4 x^{5}-x^{4}+6 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 148\)
\(\displaystyle a \! \left(n \right) = 2 a \! \left(n +2\right)+\frac{a \! \left(n +3\right)}{2}-3 a \! \left(n +4\right)+4 a \! \left(n +5\right)-\frac{5 a \! \left(n +6\right)}{2}+\frac{a \! \left(n +7\right)}{2}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{71499780 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +5}}{369101377}+\frac{71499780 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +5}}{369101377}+\frac{71499780 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +5}}{369101377}+\frac{71499780 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +5}}{369101377}+\frac{71499780 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +5}}{369101377}+\frac{71499780 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +5}}{369101377}+\frac{71499780 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +5}}{369101377}+\frac{34721510 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +4}}{369101377}+\frac{34721510 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +4}}{369101377}+\frac{34721510 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +4}}{369101377}+\frac{34721510 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +4}}{369101377}+\frac{34721510 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +4}}{369101377}+\frac{34721510 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +4}}{369101377}+\frac{34721510 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +4}}{369101377}-\frac{139278976 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +3}}{369101377}-\frac{139278976 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +3}}{369101377}-\frac{139278976 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +3}}{369101377}-\frac{139278976 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +3}}{369101377}-\frac{139278976 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +3}}{369101377}-\frac{139278976 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +3}}{369101377}-\frac{139278976 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +3}}{369101377}-\frac{117675096 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +2}}{369101377}-\frac{117675096 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +2}}{369101377}-\frac{117675096 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +2}}{369101377}-\frac{117675096 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +2}}{369101377}-\frac{117675096 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +2}}{369101377}-\frac{117675096 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +2}}{369101377}-\frac{117675096 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +2}}{369101377}+\frac{168764689 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +1}}{369101377}+\frac{168764689 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +1}}{369101377}+\frac{168764689 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +1}}{369101377}+\frac{168764689 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +1}}{369101377}+\frac{168764689 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +1}}{369101377}+\frac{168764689 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +1}}{369101377}+\frac{168764689 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +1}}{369101377}+\frac{75023957 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n -1}}{369101377}+\frac{75023957 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n -1}}{369101377}+\frac{75023957 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n -1}}{369101377}+\frac{75023957 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n -1}}{369101377}+\frac{75023957 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n -1}}{369101377}+\frac{75023957 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n -1}}{369101377}+\frac{75023957 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n -1}}{369101377}-\frac{139286860 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n}}{369101377}-\frac{139286860 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n}}{369101377}-\frac{139286860 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n}}{369101377}-\frac{139286860 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n}}{369101377}-\frac{139286860 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n}}{369101377}-\frac{139286860 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n}}{369101377}-\frac{139286860 \mathit{RootOf} \left(2 Z^{7}-4 Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n}}{369101377}\)
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This specification was found using the strategy pack "Point Placements" and has 93 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{20}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \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_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{53}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{54}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{86}\! \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_{84}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{37}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{74}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{44}\! \left(x \right)\\ \end{align*}\)