Av(1234, 1243, 1324, 2314, 3124)
Generating Function
\(\displaystyle \frac{x^{2}+1-\sqrt{x^{4}-2 x^{2}-4 x +1}}{2 \left(x +1\right) x}\)
Counting Sequence
1, 1, 2, 6, 19, 64, 225, 816, 3031, 11473, 44096, 171631, 675130, 2679728, 10719237, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x +1\right) x F \left(x
\right)^{2}+\left(-x^{2}-1\right) F \! \left(x \right)+1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -\frac{n a \! \left(n \right)}{6+n}-\frac{n a \! \left(1+n \right)}{6+n}+\frac{2 \left(n +3\right) a \! \left(n +2\right)}{6+n}+\frac{6 \left(n +4\right) a \! \left(n +3\right)}{6+n}+\frac{3 \left(n +4\right) a \! \left(n +4\right)}{6+n}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -\frac{n a \! \left(n \right)}{6+n}-\frac{n a \! \left(1+n \right)}{6+n}+\frac{2 \left(n +3\right) a \! \left(n +2\right)}{6+n}+\frac{6 \left(n +4\right) a \! \left(n +3\right)}{6+n}+\frac{3 \left(n +4\right) a \! \left(n +4\right)}{6+n}, \quad n \geq 5\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 18 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 18 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
\end{align*}\)