Av(12354, 12453, 13254, 13452, 14253, 14352, 21354, 21453, 23154, 24153, 31254, 32154)
Generating Function
\(\displaystyle \frac{2 x^{2}-4 x -1+\sqrt{4 x^{4}+12 x^{2}-8 x +1}}{8 x \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 108, 512, 2504, 12528, 63824, 330016, 1727712, 9140352, 48792256, 262485760, ...
Implicit Equation for the Generating Function
\(\displaystyle 4 x \left(x -1\right) F \left(x
\right)^{2}+\left(-2 x^{2}+4 x +1\right) F \! \left(x \right)-x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 5 \right)} = \frac{4 n a{\left(n \right)}}{n + 6} - \frac{4 n a{\left(n + 1 \right)}}{n + 6} + \frac{12 \left(n + 3\right) a{\left(n + 2 \right)}}{n + 6} + \frac{3 \left(3 n + 14\right) a{\left(n + 4 \right)}}{n + 6} - \frac{4 \left(5 n + 18\right) a{\left(n + 3 \right)}}{n + 6}, \quad n \geq 5\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 5 \right)} = \frac{4 n a{\left(n \right)}}{n + 6} - \frac{4 n a{\left(n + 1 \right)}}{n + 6} + \frac{12 \left(n + 3\right) a{\left(n + 2 \right)}}{n + 6} + \frac{3 \left(3 n + 14\right) a{\left(n + 4 \right)}}{n + 6} - \frac{4 \left(5 n + 18\right) a{\left(n + 3 \right)}}{n + 6}, \quad n \geq 5\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 23 rules.
Finding the specification took 5840 seconds.
Copy 23 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= -F_{13}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{19}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= \frac{F_{18}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{18}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 23 rules.
Finding the specification took 2633 seconds.
Copy 23 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{13}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{0}\! \left(x \right) F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{13}\! \left(x \right)}\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{21}\! \left(x \right) &= -F_{22}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)\\
\end{align*}\)