Av(1342, 1432, 3214)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{4}}{2 x^{6}-8 x^{5}+19 x^{4}-27 x^{3}+19 x^{2}-7 x +1}\)
Counting Sequence
1, 1, 2, 6, 21, 74, 249, 804, 2540, 7977, 25106, 79327, 251328, 797094, 2527977, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{6}-8 x^{5}+19 x^{4}-27 x^{3}+19 x^{2}-7 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(n +6\right) = -2 a \! \left(n \right)+8 a \! \left(n +1\right)-19 a \! \left(n +2\right)+27 a \! \left(n +3\right)-19 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(n +6\right) = -2 a \! \left(n \right)+8 a \! \left(n +1\right)-19 a \! \left(n +2\right)+27 a \! \left(n +3\right)-19 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{11060 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +4}}{954551}-\frac{11060 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +4}}{954551}-\frac{11060 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +4}}{954551}-\frac{11060 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +4}}{954551}-\frac{11060 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +4}}{954551}-\frac{11060 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +4}}{954551}+\frac{33880 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +3}}{954551}+\frac{33880 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +3}}{954551}+\frac{33880 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +3}}{954551}+\frac{33880 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +3}}{954551}+\frac{33880 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +3}}{954551}+\frac{33880 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +3}}{954551}-\frac{121666 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +2}}{954551}-\frac{121666 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +2}}{954551}-\frac{121666 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +2}}{954551}-\frac{121666 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +2}}{954551}-\frac{121666 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +2}}{954551}-\frac{121666 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +2}}{954551}+\frac{95759 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +1}}{954551}+\frac{95759 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +1}}{954551}+\frac{95759 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +1}}{954551}+\frac{95759 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +1}}{954551}+\frac{95759 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +1}}{954551}+\frac{95759 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +1}}{954551}+\frac{57501 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n -1}}{954551}+\frac{57501 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n -1}}{954551}+\frac{57501 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n -1}}{954551}+\frac{57501 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n -1}}{954551}+\frac{57501 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n -1}}{954551}+\frac{57501 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n -1}}{954551}+\frac{32960 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n}}{954551}+\frac{32960 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n}}{954551}+\frac{32960 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n}}{954551}+\frac{32960 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n}}{954551}+\frac{32960 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n}}{954551}+\frac{32960 \mathit{RootOf} \left(2 Z^{6}-8 Z^{5}+19 Z^{4}-27 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n}}{954551}\)
This specification was found using the strategy pack "Point Placements" and has 94 rules.
Found on January 18, 2022.Finding the specification took 4 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 94 equations to clipboard:
\(\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_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \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_{14}\! \left(x \right)+F_{7}\! \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_{80}\! \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_{35}\! \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_{26}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \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_{32}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \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_{14}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{54}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{74}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{90}\! \left(x \right)\\
\end{align*}\)