Av(1243, 1342, 2314)
Generating Function
\(\displaystyle -\frac{4 \left(\left(x -\frac{1}{2}\right) \sqrt{-4 x +1}+x^{3}-4 x^{2}+3 x -\frac{1}{2}\right) \left(x -\frac{1}{2}\right)}{2 x^{5}-16 x^{4}+44 x^{3}-42 x^{2}+16 x -2}\)
Counting Sequence
1, 1, 2, 6, 21, 77, 286, 1067, 3993, 14992, 56488, 213600, 810449, 3084733, 11774727, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-8 x^{4}+22 x^{3}-21 x^{2}+8 x -1\right) F \left(x
\right)^{2}+\left(2 x -1\right) \left(2 x^{3}-8 x^{2}+6 x -1\right) F \! \left(x \right)+x \left(2 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 286\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(5+2 n \right) a \! \left(n \right)}{n +7}-\frac{10 \left(19+7 n \right) a \! \left(n +1\right)}{n +7}+\frac{3 \left(229+75 n \right) a \! \left(n +2\right)}{n +7}-\frac{4 \left(284+77 n \right) a \! \left(n +3\right)}{n +7}+\frac{4 \left(236+53 n \right) a \! \left(n +4\right)}{n +7}-\frac{11 \left(37+7 n \right) a \! \left(n +5\right)}{n +7}+\frac{2 \left(43+7 n \right) a \! \left(n +6\right)}{n +7}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 286\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(5+2 n \right) a \! \left(n \right)}{n +7}-\frac{10 \left(19+7 n \right) a \! \left(n +1\right)}{n +7}+\frac{3 \left(229+75 n \right) a \! \left(n +2\right)}{n +7}-\frac{4 \left(284+77 n \right) a \! \left(n +3\right)}{n +7}+\frac{4 \left(236+53 n \right) a \! \left(n +4\right)}{n +7}-\frac{11 \left(37+7 n \right) a \! \left(n +5\right)}{n +7}+\frac{2 \left(43+7 n \right) a \! \left(n +6\right)}{n +7}, \quad n \geq 7\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 30 rules.
Found on July 23, 2021.Finding the specification took 6 seconds.
Copy 30 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_{23}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{23}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\
F_{7}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{23}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{23}\! \left(x \right)\\
F_{17}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{4}\! \left(x \right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= -\frac{-y F_{15}\! \left(x , y\right)+F_{15}\! \left(x , 1\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= y x\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{23}\! \left(x \right)\\
F_{22}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{23}\! \left(x \right) &= x\\
F_{24}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{26}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
F_{27}\! \left(x \right) &= F_{23}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\
F_{29}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\
\end{align*}\)