Av(13452, 13524, 13542, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 31452, 31524, 31542, 35124, 35142, 35214, 51234, 51243, 51324, 51342, 51423, 51432, 52134, 52143, 52314, 53124, 53142, 53214)
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Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(6 x^{4}+8 x^{3}+2 x -1\right)}{6 x^{5}+14 x^{4}+8 x^{3}+2 x^{2}+2 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 90, 310, 1088, 3888, 13836, 49032, 173932, 617576, 2192304, 7780680, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(6 x^{5}+14 x^{4}+8 x^{3}+2 x^{2}+2 x -1\right) F \! \left(x \right)-\left(x +1\right) \left(6 x^{4}+8 x^{3}+2 x -1\right) = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a{\left(n + 5 \right)} = 6 a{\left(n \right)} + 14 a{\left(n + 1 \right)} + 8 a{\left(n + 2 \right)} + 2 a{\left(n + 3 \right)} + 2 a{\left(n + 4 \right)}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{53 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +1}}{149}\\+\\\frac{830 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +2}}{2831}\\-\\\frac{4398 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +3}}{2831}\\-\\\frac{4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +4}}{2831}\\+\\\frac{\left(4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{3}+10920 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}+6240 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+553\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +1}}{2831}\\+\\\frac{\left(4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}+10920 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+7070\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +2}}{2831}\\+\\\frac{\left(4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+6522\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +3}}{2831}\\+\\\frac{\left(\left(-4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)-6522\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)^{2}+\left(-4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}-17442 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)-15218\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)-6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}-15218 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)-8143\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +1}}{2831}\\+\\\frac{\left(\left(-4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)-6522\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)-6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)-8148\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +2}}{2831}\\+\\\frac{\left(\left(\left(4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+6522\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+8148\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)+\left(6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+8148\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+8148 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+10869\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +1}}{2831}\\+\\\frac{783 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n}}{2831}\\+\\\frac{\left(4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{4}+10920 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{3}+6240 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}+1560 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+2343\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n}}{2831}\\+\\\frac{\left(\left(-4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)-6522\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)^{3}+\left(-4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}-17442 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)-15218\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)^{2}+\left(-4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{3}-17442 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}-21458 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)-8696\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)-6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{3}-15218 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}-8696 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+169\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n}}{2831}\\+\\\frac{\left(\left(\left(4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+6522\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+8148\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)^{2}+\left(\left(4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+6522\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}+\left(4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)^{2}+23964 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+23366\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)^{2}+23366 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+19012\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)+\left(6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+8148\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)^{2}+\left(6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)^{2}+23366 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+19012\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+8148 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)^{2}+19012 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)+11033\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n}}{2831}\\+\\\frac{\left(\left(\left(\left(-4680 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)-6522\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)-6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)-8148\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+\left(-6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)-8148\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)-8148 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)-10869\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =2\right)+\left(\left(-6522 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)-8148\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)-8148 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)-10869\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =1\right)+\left(-8148 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)-10869\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =3\right)-10869 \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =4\right)-14328\right) \mathit{RootOf}\left(6 Z^{5}+14 Z^{4}+8 Z^{3}+2 Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n}}{2831} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 51 rules.

Finding the specification took 91 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_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{7}\! \left(x \right) &= 0\\ F_{8}\! \left(x \right) &= F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{20}\! \left(x \right) &= x\\ F_{21}\! \left(x \right) &= F_{20}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{27}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{20}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{20}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{33}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{20}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{38}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{20}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{20}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{46}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{20}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{38}\! \left(x \right)+F_{7}\! \left(x \right)\\ \end{align*}\)