Av(1234, 1243, 1324, 1342, 2134, 3124)
Generating Function
\(\displaystyle \frac{-\left(x^{2}+1\right)^{2} \sqrt{-4 x +1}+3 x^{4}+4 x^{2}-2 x +1}{2 \left(x^{3}+2 x^{2}+2 x +2\right) x^{2}}\)
Counting Sequence
1, 1, 2, 6, 18, 56, 182, 607, 2064, 7132, 24970, 88383, 315748, 1137014, 4122762, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+2 x^{2}+2 x +2\right) x^{2} F \left(x
\right)^{2}+\left(-3 x^{4}-4 x^{2}+2 x -1\right) F \! \left(x \right)+x^{4}+2 x^{2}-x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(n +6\right) = \frac{\left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{\left(13+5 n \right) a \! \left(2+n \right)}{8+n}+\frac{\left(2+7 n \right) a \! \left(n +1\right)}{16+2 n}+\frac{\left(44+13 n \right) a \! \left(n +3\right)}{16+2 n}+2 a \! \left(n +4\right)+\frac{3 \left(n +6\right) a \! \left(n +5\right)}{8+n}, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(n +6\right) = \frac{\left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{\left(13+5 n \right) a \! \left(2+n \right)}{8+n}+\frac{\left(2+7 n \right) a \! \left(n +1\right)}{16+2 n}+\frac{\left(44+13 n \right) a \! \left(n +3\right)}{16+2 n}+2 a \! \left(n +4\right)+\frac{3 \left(n +6\right) a \! \left(n +5\right)}{8+n}, \quad n \geq 6\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 23 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 23 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= x\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= y x\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{7}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{21}\! \left(x , y\right) &= \frac{y F_{13}\! \left(x , y\right)-F_{13}\! \left(x , 1\right)}{-1+y}\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\
\end{align*}\)