Av(12435, 12453, 21435, 21453)
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
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{8 \left(2 n + 1\right) a{\left(n \right)}}{n + 2} - \frac{6 \left(3 n + 2\right) a{\left(n + 1 \right)}}{n + 2} + \frac{2 \left(5 n + 4\right) a{\left(n + 2 \right)}}{n + 2}, \quad n \geq 3\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{8 \left(2 n + 1\right) a{\left(n \right)}}{n + 2} - \frac{6 \left(3 n + 2\right) a{\left(n + 1 \right)}}{n + 2} + \frac{2 \left(5 n + 4\right) a{\left(n + 2 \right)}}{n + 2}, \quad n \geq 3\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 22 rules.
Finding the specification took 56 seconds.
Copy 22 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= x F_{6}\! \left(x , y\right) y +F_{6}\! \left(x , y\right)^{2}-2 F_{6}\! \left(x , y\right)+2\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{11}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{6}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= x F_{15}\! \left(x , y\right) y +y x +F_{15}\! \left(x , y\right)^{2}\\
F_{16}\! \left(x , y\right) &= y x\\
F_{17}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\
F_{19}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\
F_{5}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{20}\! \left(x , y\right)\\
F_{21}\! \left(x \right) &= x\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 22 rules.
Finding the specification took 70 seconds.
Copy 22 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= x F_{7}\! \left(x , y\right) y +y x +F_{7}\! \left(x , y\right)^{2}\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{21}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= y x\\
F_{18}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)\\
F_{21}\! \left(x \right) &= x\\
\end{align*}\)