Av(1234, 1342, 1423, 2314, 2341)
Generating Function
\(\displaystyle \frac{\left(x -1\right) \sqrt{-4 x +1}-2 x^{2}-x +1}{2 x \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 60, 191, 619, 2048, 6909, 23704, 82489, 290500, 1033399, 3707838, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(x +1\right) \left(2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{3}+2 x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +2\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{3+n}+\frac{\left(9+5 n \right) a \! \left(1+n \right)}{3+n}+\frac{3+3 n}{3+n}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +2\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{3+n}+\frac{\left(9+5 n \right) a \! \left(1+n \right)}{3+n}+\frac{3+3 n}{3+n}, \quad n \geq 4\)
This specification was found using the strategy pack "Row And Col Placements Req Corrob Expand Verified" and has 28 rules.
Found on January 21, 2022.Finding the specification took 21 seconds.
Copy 28 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_{8}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{6}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\
F_{11}\! \left(x , y\right) &= \frac{y F_{12}\! \left(x , y\right)-F_{12}\! \left(x , 1\right)}{-1+y}\\
F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= y x\\
F_{15}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{8}\! \left(x \right)}\\
F_{20}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{8}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25} \left(x \right)^{2} F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{25}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row And Col Placements Req Corrob Expand Verified" and has 29 rules.
Found on January 21, 2022.Finding the specification took 29 seconds.
Copy 29 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_{18}\! \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_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\
F_{13}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= y x\\
F_{17}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{13}\! \left(x , y\right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{10}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= \frac{F_{6}\! \left(x \right)}{F_{10}\! \left(x \right)}\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26} \left(x \right)^{2} F_{10}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row Placements Req Corrob Expand Verified" and has 35 rules.
Found on January 21, 2022.Finding the specification took 35 seconds.
Copy 35 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_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{17}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= -F_{33}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{10}\! \left(x \right) &= \frac{F_{11}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{11}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{25}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x \right) &= F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\
F_{20}\! \left(x , y\right) &= -\frac{-y F_{21}\! \left(x , y\right)+F_{21}\! \left(x , 1\right)}{-1+y}\\
F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= y x\\
F_{24}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{20}\! \left(x , y\right)\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{28}\! \left(x \right) &= 0\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{17}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{12}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{17}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{17}\! \left(x \right) F_{33}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row And Col Placements Expand Verified" and has 29 rules.
Found on January 21, 2022.Finding the specification took 24 seconds.
Copy 29 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_{9}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{17}\! \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_{9}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\
F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\
F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= y x\\
F_{16}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{9}\! \left(x \right)}\\
F_{21}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{9}\! \left(x \right)}\\
F_{23}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26} \left(x \right)^{2} F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{9}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Expand Verified" and has 30 rules.
Found on January 21, 2022.Finding the specification took 32 seconds.
Copy 30 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_{18}\! \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_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\
F_{13}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= y x\\
F_{17}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{13}\! \left(x , y\right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{10}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{6}\! \left(x \right)}{F_{10}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27} \left(x \right)^{2} F_{10}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right) F_{27}\! \left(x \right)\\
\end{align*}\)