Av(1234, 1324, 1342, 1423, 2314)
Generating Function
\(\displaystyle \frac{\left(-2 x^{2}+2 x -1+\sqrt{-4 x +1}\right) \left(x -1\right)}{2 x^{2} \left(x^{2}-2 x +2\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 61, 200, 670, 2286, 7918, 27770, 98424, 351983, 1268541, 4602752, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x^{2}-2 x +2\right) F \left(x
\right)^{2}+\left(x -1\right) \left(2 x^{2}-2 x +1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{\left(5+2 n \right) a \! \left(n \right)}{6+n}-\frac{\left(70+19 n \right) a \! \left(2+n \right)}{2 \left(6+n \right)}+\frac{\left(42+13 n \right) a \! \left(n +1\right)}{12+2 n}+\frac{2 \left(14+3 n \right) a \! \left(n +3\right)}{6+n}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{\left(5+2 n \right) a \! \left(n \right)}{6+n}-\frac{\left(70+19 n \right) a \! \left(2+n \right)}{2 \left(6+n \right)}+\frac{\left(42+13 n \right) a \! \left(n +1\right)}{12+2 n}+\frac{2 \left(14+3 n \right) a \! \left(n +3\right)}{6+n}, \quad n \geq 4\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 20 rules.
Found on July 23, 2021.Finding the specification took 21 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\
\end{align*}\)