Av(1243, 1432, 2314, 3142)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x -1\right)^{5}}{5 x^{6}-26 x^{5}+47 x^{4}-46 x^{3}+26 x^{2}-8 x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 197, 597, 1804, 5454, 16495, 49883, 150827, 456009, 1378707, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(5 x^{6}-26 x^{5}+47 x^{4}-46 x^{3}+26 x^{2}-8 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{5} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 197\)
\(\displaystyle a \! \left(n +6\right) = -5 a \! \left(n \right)+26 a \! \left(n +1\right)-47 a \! \left(n +2\right)+46 a \! \left(n +3\right)-26 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle -\frac{115 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +4}}{181}-\frac{115 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +4}}{181}+\frac{2447 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +3}}{724}+\frac{2447 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +3}}{724}-\frac{2205 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +2}}{362}-\frac{2205 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +2}}{362}+\frac{3811 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +1}}{724}+\frac{3811 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +1}}{724}+\frac{333 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n -1}}{724}+\frac{333 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n -1}}{724}-\frac{1681 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n}}{724}-\frac{1681 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n}}{724}-\frac{1681 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n}}{724}-\frac{1681 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n}}{724}-\frac{1681 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n}}{724}-\frac{1681 \mathit{RootOf} \left(5 Z^{6}-26 Z^{5}+47 Z^{4}-46 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n}}{724}+\left(\left\{\begin{array}{cc}\frac{2}{5} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Point Placements" and has 64 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{12}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\ \end{align*}\)