Av(12345, 12354, 21345, 21354)
Generating Function
\(\displaystyle \frac{-x \sqrt{-8 x +1}-x +2}{4 x^{2}-4 x +2}\)
Counting Sequence
1, 1, 2, 6, 24, 116, 632, 3720, 23072, 148528, 983072, 6647776, 45727616, 318947136, 2250473344, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{2}-2 x +1\right) F \left(x
\right)^{2}+\left(x -2\right) F \! \left(x \right)+x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{6 \left(3 n +2\right) a \! \left(n +1\right)}{n +2}+\frac{2 \left(5 n +4\right) a \! \left(n +2\right)}{n +2}, \quad n \geq 3\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{6 \left(3 n +2\right) a \! \left(n +1\right)}{n +2}+\frac{2 \left(5 n +4\right) a \! \left(n +2\right)}{n +2}, \quad n \geq 3\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 22 rules.
Found on January 23, 2022.Finding the specification took 194 seconds.
Copy 22 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{18}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{18}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{19}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= y x\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{18}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{5}\! \left(x \right)\\
F_{17}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right) F_{5}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 23 rules.
Found on January 22, 2022.Finding the specification took 11 seconds.
Copy 23 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_{17}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , 1, y\right)\\
F_{9}\! \left(x , y , z\right) &= F_{10}\! \left(x , y z , z\right)\\
F_{10}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , z , y\right)+F_{13}\! \left(x , y , z\right)+F_{15}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)\\
F_{11}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{6}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{13}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{14}\! \left(x , y , z\right)\\
F_{14}\! \left(x , y , z\right) &= -\frac{z F_{9}\! \left(x , 1, z\right)-y F_{9}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right) F_{17}\! \left(x \right)\\
F_{16}\! \left(x , y , z\right) &= -\frac{-z F_{10}\! \left(x , y , z\right)+F_{10}\! \left(x , y , 1\right)}{-1+z}\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right) F_{17}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{21}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{20}\! \left(x , y\right)\\
F_{22}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\
\end{align*}\)