Av(1234, 1342, 2413)
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Generating Function
\(\displaystyle -\frac{\left(3 x -1\right) \left(x -1\right)^{4}}{2 x^{6}-10 x^{5}+28 x^{4}-36 x^{3}+24 x^{2}-8 x +1}\)
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Counting Sequence
1, 1, 2, 6, 21, 75, 264, 918, 3176, 10978, 37964, 131362, 454692, 1574092, 5449596, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{6}-10 x^{5}+28 x^{4}-36 x^{3}+24 x^{2}-8 x +1\right) F \! \left(x \right)+\left(3 x -1\right) \left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(n +6\right) = -2 a \! \left(n \right)+10 a \! \left(n +1\right)-28 a \! \left(n +2\right)+36 a \! \left(n +3\right)-24 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle -\frac{2228 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +4}}{12469}-\frac{2228 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +4}}{12469}+\frac{9450 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +3}}{12469}+\frac{9450 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +3}}{12469}-\frac{24203 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +2}}{12469}-\frac{24203 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +2}}{12469}+\frac{21667 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +1}}{12469}+\frac{21667 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +1}}{12469}+\frac{2893 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n -1}}{24938}+\frac{2893 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n -1}}{24938}-\frac{15095 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n}}{24938}-\frac{15095 \mathit{RootOf} \left(2 Z^{6}-10 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n}}{24938}\)
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This specification was found using the strategy pack "Point Placements" and has 73 rules.

Found on January 18, 2022.

Finding the specification took 7 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_{64}\! \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_{34}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \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_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \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_{33}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \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_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{54}\! \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_{48}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{12}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{70}\! \left(x \right)\\ \end{align*}\)