Av(13425, 13452, 14235, 14253, 14325, 14352, 14523, 14532)
Counting Sequence
1, 1, 2, 6, 24, 112, 570, 3066, 17142, 98622, 580038, 3471690, 21077100, 129482964, 803417270, ...
Implicit Equation for the Generating Function
\(\displaystyle F \left(x
\right)^{3}-3 F \left(x
\right)^{2}+\left(3 x^{2}-2 x +3\right) F \! \left(x \right)-\left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = \frac{81 \left(n -1\right) n a \! \left(n \right)}{4 \left(2 n +7\right) \left(n +3\right)}-\frac{27 n \left(5 n +1\right) a \! \left(n +1\right)}{4 \left(2 n +7\right) \left(n +3\right)}+\frac{9 \left(15 n^{2}+21 n +2\right) a \! \left(n +2\right)}{4 \left(2 n +7\right) \left(n +3\right)}+\frac{3 \left(13 n^{2}+77 n +110\right) a \! \left(n +3\right)}{4 \left(2 n +7\right) \left(n +3\right)}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = \frac{81 \left(n -1\right) n a \! \left(n \right)}{4 \left(2 n +7\right) \left(n +3\right)}-\frac{27 n \left(5 n +1\right) a \! \left(n +1\right)}{4 \left(2 n +7\right) \left(n +3\right)}+\frac{9 \left(15 n^{2}+21 n +2\right) a \! \left(n +2\right)}{4 \left(2 n +7\right) \left(n +3\right)}+\frac{3 \left(13 n^{2}+77 n +110\right) a \! \left(n +3\right)}{4 \left(2 n +7\right) \left(n +3\right)}, \quad n \geq 4\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 20 rules.
Found on January 23, 2022.Finding the specification took 84 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)^{2} F_{12}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{19}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{6}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{19}\! \left(x \right) &= x\\
\end{align*}\)