Av(12354, 12453, 13254, 13452, 14253, 14352, 15243, 15342, 23451, 24351, 25341)
Generating Function
\(\displaystyle \frac{-\sqrt{6 x -1}\, \left(2 x -1\right)^{\frac{3}{2}}+6 x^{3}-10 x^{2}+2 x +1}{2 x \left(3 x -2\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 109, 522, 2574, 12964, 66426, 345300, 1816976, 9660732, 51825093, 280168474, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(3 x -2\right)^{2} F \left(x
\right)^{2}+\left(-6 x^{3}+10 x^{2}-2 x -1\right) F \! \left(x \right)+x^{3}-2 x^{2}-x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = \frac{18 \left(1+n \right) a \! \left(n \right)}{n +4}-\frac{3 \left(13+8 n \right) a \! \left(1+n \right)}{n +4}+\frac{\left(51+19 n \right) a \! \left(n +2\right)}{2 n +8}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = \frac{18 \left(1+n \right) a \! \left(n \right)}{n +4}-\frac{3 \left(13+8 n \right) a \! \left(1+n \right)}{n +4}+\frac{\left(51+19 n \right) a \! \left(n +2\right)}{2 n +8}, \quad n \geq 4\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 29 rules.
Found on January 23, 2022.Finding the specification took 167 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_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{28}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= y x\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{28}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= \frac{F_{20}\! \left(x , y\right) y -F_{20}\! \left(x , 1\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{28}\! \left(x \right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{26}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{21}\! \left(x , 1\right)\\
F_{28}\! \left(x \right) &= x\\
\end{align*}\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 20 rules.
Found on January 22, 2022.Finding the specification took 8 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{6}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x \right) &= x\\
F_{8}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= \frac{y F_{10}\! \left(x , 1, y\right)-F_{10}\! \left(x , \frac{1}{y}, y\right)}{-1+y}\\
F_{10}\! \left(x , y , z\right) &= F_{11}\! \left(x , y z , z\right)\\
F_{11}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)\\
F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x , y , z\right)\\
F_{13}\! \left(x , y , z\right) &= F_{11}\! \left(x , y , z\right) F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= y x\\
F_{18}\! \left(x , y , z\right) &= F_{17}\! \left(x , z\right) F_{19}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= -\frac{z F_{10}\! \left(x , 1, z\right)-y F_{10}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
\end{align*}\)