Av(1234, 1342, 2314, 2413)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{3}}{x^{5}-6 x^{4}+14 x^{3}-13 x^{2}+6 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 203, 619, 1871, 5651, 17096, 51796, 157035, 476171, 1443790, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-6 x^{4}+14 x^{3}-13 x^{2}+6 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-6 a \! \left(n +1\right)+14 a \! \left(n +2\right)-13 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle -\frac{272 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{4417}-\frac{272 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{4417}-\frac{272 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3}}{4417}-\frac{272 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3}}{4417}-\frac{272 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3}}{4417}+\frac{1026 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{4417}+\frac{1026 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{4417}+\frac{1026 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{4417}+\frac{1026 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2}}{4417}+\frac{1026 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2}}{4417}-\frac{1717 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{4417}-\frac{1717 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{4417}-\frac{1717 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{4417}-\frac{1717 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{4417}-\frac{1717 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1}}{4417}+\frac{82 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n -1}}{4417}+\frac{82 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n -1}}{4417}+\frac{82 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n -1}}{4417}+\frac{82 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n -1}}{4417}+\frac{82 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n -1}}{4417}+\frac{1367 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{4417}+\frac{1367 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{4417}+\frac{1367 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{4417}+\frac{1367 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{4417}+\frac{1367 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{4417}\)
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This specification was found using the strategy pack "Point Placements" and has 44 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_{21}\! \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_{14}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= 0\\ F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{41}\! \left(x \right)\\ \end{align*}\)