Av(12453, 12543, 21453, 21543, 31452, 31542, 41352, 41532, 51342, 51432)
Generating Function
\(\displaystyle \frac{\left(-2 x^{3}+9 x^{2}-x \right) \sqrt{x^{2}-6 x +1}-2 x^{4}+15 x^{3}-18 x^{2}-9 x +2}{2 x^{2}-12 x +2}\)
Counting Sequence
1, 1, 2, 6, 24, 110, 540, 2758, 14448, 77022, 415860, 2267078, 12452616, 68814798, 382168332, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-6 x +1\right) F \left(x
\right)^{2}+\left(x -2\right) \left(2 x +1\right) \left(x^{2}-6 x +1\right) F \! \left(x \right)-2 x^{4}+3 x^{3}-17 x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(n -2\right) a \! \left(n \right)}{3+n}+\frac{3 \left(8+5 n \right) a \! \left(3+n \right)}{3+n}+\frac{3 \left(-6+7 n \right) a \! \left(n +1\right)}{3+n}-\frac{3 \left(9+19 n \right) a \! \left(n +2\right)}{3+n}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(n -2\right) a \! \left(n \right)}{3+n}+\frac{3 \left(8+5 n \right) a \! \left(3+n \right)}{3+n}+\frac{3 \left(-6+7 n \right) a \! \left(n +1\right)}{3+n}-\frac{3 \left(9+19 n \right) a \! \left(n +2\right)}{3+n}, \quad n \geq 5\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 18 rules.
Found on January 23, 2022.Finding the specification took 0 seconds.
Copy 18 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= \frac{y F_{7}\! \left(x , y\right)-F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= y x\\
\end{align*}\)