Av(1234, 1243, 1432, 2143)
Generating Function
\(\displaystyle \frac{x^{4}-2 x^{3}-2 x^{2}-2 x +1}{2 x^{4}-x^{3}-x^{2}-3 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 220, 734, 2448, 8166, 27240, 90866, 303108, 1011098, 3372788, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}-x^{3}-x^{2}-3 x +1\right) F \! \left(x \right)-x^{4}+2 x^{3}+2 x^{2}+2 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+3 a \! \left(n +3\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+3 a \! \left(n +3\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{368 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-Z^{3}-Z^{2}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{4155}+\frac{219 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-Z^{3}-Z^{2}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{1385}+\frac{58 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-Z^{3}-Z^{2}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{831}+\frac{43 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-Z^{3}-Z^{2}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{4155}+\left(\left\{\begin{array}{cc}\frac{1}{2} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 71 rules.
Found on January 18, 2022.Finding the specification took 6 seconds.
Copy 71 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_{19}\! \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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)+F_{67}\! \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_{64}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{68}\! \left(x \right)\\
\end{align*}\)