Av(1342, 1423, 1432, 2314)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x^{2}+2 x -1\right)}{\left(x -1\right) \left(2 x^{3}+2 x^{2}-4 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 214, 686, 2186, 6946, 22042, 69906, 221650, 702706, 2227714, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x^{3}+2 x^{2}-4 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x^{2}+2 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = -2 a \! \left(n \right)-2 a \! \left(n +1\right)+4 a \! \left(n +2\right)+2, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = -2 a \! \left(n \right)-2 a \! \left(n +1\right)+4 a \! \left(n +2\right)+2, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \frac{270 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-6 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{37}+\frac{239 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-6 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{37}-16 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-6 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)+\frac{157 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-6 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{37}\)
This specification was found using the strategy pack "Point Placements" and has 49 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 49 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{13}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{10}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{10}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= x^{2}\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{10}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{10}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\
\end{align*}\)