Av(1342, 1432, 2314)
Generating Function
\(\displaystyle -\frac{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}{\left(x -1\right) \left(x^{4}-13 x^{3}+16 x^{2}-7 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 264, 914, 3127, 10621, 35932, 121324, 409301, 1380417, 4655382, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{4}-13 x^{3}+16 x^{2}-7 x +1\right) F \! \left(x \right)+\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+13 a \! \left(n +1\right)-16 a \! \left(n +2\right)+7 a \! \left(n +3\right)-1, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+13 a \! \left(n +1\right)-16 a \! \left(n +2\right)+7 a \! \left(n +3\right)-1, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{619 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-14 Z^{4}+29 Z^{3}-23 Z^{2}+8 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{1192}+\frac{8187 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-14 Z^{4}+29 Z^{3}-23 Z^{2}+8 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{1192}-\frac{5797 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-14 Z^{4}+29 Z^{3}-23 Z^{2}+8 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{596}+\frac{5323 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-14 Z^{4}+29 Z^{3}-23 Z^{2}+8 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{1192}-\frac{701 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-14 Z^{4}+29 Z^{3}-23 Z^{2}+8 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{1192}\)
This specification was found using the strategy pack "Point Placements" and has 61 rules.
Found on January 18, 2022.Finding the specification took 2 seconds.
Copy 61 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_{16}\! \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_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= 0\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \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_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{10}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{36}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{10}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \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_{40}\! \left(x \right) &= x^{2}\\
F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{10}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{10}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{49}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{32}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{10}\! \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_{25}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{10}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{50}\! \left(x \right)+F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{10}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{56}\! \left(x \right)\\
\end{align*}\)