Av(1234, 1342, 1432, 2143)
Generating Function
\(\displaystyle \frac{x^{5}+2 x^{4}-x^{3}-2 x^{2}-2 x +1}{\left(x -1\right) \left(x^{4}+3 x^{3}+3 x^{2}+2 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 207, 671, 2174, 7045, 22831, 73989, 239779, 777062, 2518258, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{4}+3 x^{3}+3 x^{2}+2 x -1\right) F \! \left(x \right)-x^{5}-2 x^{4}+x^{3}+2 x^{2}+2 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+3 a \! \left(n +1\right)+3 a \! \left(n +2\right)+2 a \! \left(n +3\right)-1, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+3 a \! \left(n +1\right)+3 a \! \left(n +2\right)+2 a \! \left(n +3\right)-1, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{1833 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{44768}-\frac{263 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{44768}+\frac{6069 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{44768}-\frac{211 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{22384}+\frac{2045 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{44768}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 87 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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= x^{2}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{61}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{55}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{40}\! \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_{55}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{61}\! \left(x \right)+F_{70}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{76}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{56}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{56}\! \left(x \right)\\
\end{align*}\)