Av(1234, 1243, 1324, 2134, 14523, 34125, 351624, 356124, 451623, 456123)
Generating Function
\(\displaystyle \frac{-4 \left(x -\frac{1}{2}\right)^{2} \sqrt{-4 x +1}-2 x^{3}+10 x^{2}-6 x +1}{2 x^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 69, 242, 858, 3068, 11050, 40052, 145996, 534888, 1968685, 7276050, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{3} F \left(x
\right)^{2}+\left(2 x^{3}-10 x^{2}+6 x -1\right) F \! \left(x \right)+x^{3}+6 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +2\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{5+n}+\frac{6 \left(3+n \right) a \! \left(n +1\right)}{5+n}, \quad n \geq 3\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +2\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{5+n}+\frac{6 \left(3+n \right) a \! \left(n +1\right)}{5+n}, \quad n \geq 3\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Expand Verified" and has 37 rules.
Found on January 23, 2022.Finding the specification took 67 seconds.
Copy 37 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_{13}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \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_{3}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= -F_{9}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= \frac{F_{8}\! \left(x \right)}{F_{13}\! \left(x \right)}\\
F_{8}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\
F_{16}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{17}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= y x\\
F_{20}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{16}\! \left(x , y\right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x , 1\right)\\
F_{26}\! \left(x , y\right) &= -\frac{-y F_{27}\! \left(x , y\right)+F_{27}\! \left(x , 1\right)}{-1+y}\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{3}\! \left(x \right)\\
F_{28}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{6}\! \left(x \right)}{F_{13}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{13}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x , 1\right)\\
F_{32}\! \left(x , y\right) &= -\frac{-y F_{33}\! \left(x , y\right)+F_{33}\! \left(x , 1\right)}{-1+y}\\
F_{33}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x , y\right)+F_{35}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{26}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{32}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion Expand Verified" and has 38 rules.
Found on January 23, 2022.Finding the specification took 86 seconds.
Copy 38 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_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{14}\! \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_{10}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= \frac{F_{9}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{9}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{10}\! \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_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\
F_{17}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\
F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= y x\\
F_{21}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{17}\! \left(x , y\right)\\
F_{22}\! \left(x \right) &= F_{14}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{14}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x , 1\right)\\
F_{27}\! \left(x , y\right) &= -\frac{-y F_{28}\! \left(x , y\right)+F_{28}\! \left(x , 1\right)}{-1+y}\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{4}\! \left(x \right)\\
F_{29}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= \frac{F_{7}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x , 1\right)\\
F_{33}\! \left(x , y\right) &= -\frac{-y F_{34}\! \left(x , y\right)+F_{34}\! \left(x , 1\right)}{-1+y}\\
F_{34}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{36}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{27}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{33}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Req Corrob Expand Verified" and has 45 rules.
Found on January 23, 2022.Finding the specification took 89 seconds.
Copy 45 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_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{44}\! \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_{17}\! \left(x \right)+F_{5}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \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_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\
F_{24}\! \left(x , y\right) &= -\frac{-y F_{25}\! \left(x , y\right)+F_{25}\! \left(x , 1\right)}{-1+y}\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{4}\! \left(x \right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{8}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x , 1\right)\\
F_{33}\! \left(x , y\right) &= -\frac{-y F_{34}\! \left(x , y\right)+F_{34}\! \left(x , 1\right)}{-1+y}\\
F_{34}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{36}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{37}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x , 1\right)\\
F_{40}\! \left(x , y\right) &= -\frac{-y F_{41}\! \left(x , y\right)+F_{41}\! \left(x , 1\right)}{-1+y}\\
F_{41}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{36}\! \left(x , y\right)+F_{42}\! \left(x , y\right)+F_{43}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{43}\! \left(x , y\right) &= F_{40}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)