Av(1234, 1342, 2413, 3214)
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Generating Function
\(\displaystyle \frac{\left(x -1\right)^{3}}{2 x^{7}-2 x^{6}+2 x^{4}+4 x^{3}-5 x^{2}+4 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 60, 166, 456, 1272, 3584, 10104, 28416, 79812, 224172, 629868, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{7}-2 x^{6}+2 x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 166\)
\(\displaystyle a \! \left(n +1\right) = a \! \left(n \right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)-\frac{5 a \! \left(n +5\right)}{2}+2 a \! \left(n +6\right)-\frac{a \! \left(n +7\right)}{2}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{716124 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +5}}{20386583}+\frac{757328 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +4}}{20386583}+\frac{240734 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +3}}{20386583}-\frac{107420 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +2}}{20386583}+\frac{1378505 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +1}}{20386583}+\frac{921663 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n -1}}{20386583}+\frac{3616444 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+2 Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n}}{20386583}\)
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This specification was found using the strategy pack "Point Placements" and has 81 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{17}\! \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_{7}\! \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_{11}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{48}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{49}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \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_{62}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{59}\! \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_{79}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{43}\! \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_{74}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{49}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{46}\! \left(x \right)\\ \end{align*}\)