Av(1243, 2431, 4132)
Generating Function
\(\displaystyle \frac{2 \left(x -\frac{1}{2}\right) \left(x -1\right)^{4} \sqrt{1-4 x}-2 x^{6}+8 x^{5}-3 x^{4}-8 x^{3}+12 x^{2}-6 x +1}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 262, 896, 3033, 10261, 34906, 119771, 415012, 1452361, 5130997, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{8} F \left(x
\right)^{2}+\left(x^{2}-3 x +1\right) \left(2 x^{6}-8 x^{5}+3 x^{4}+8 x^{3}-12 x^{2}+6 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{11}-4 x^{10}-18 x^{9}+150 x^{8}-422 x^{7}+684 x^{6}-720 x^{5}+511 x^{4}-243 x^{3}+74 x^{2}-13 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 262\)
\(\displaystyle a \! \left(7\right) = 896\)
\(\displaystyle a \! \left(8\right) = 3033\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(2 n +3\right) a \! \left(n \right)}{n +5}+\frac{2 \left(15 n +28\right) a \! \left(n +1\right)}{n +5}-\frac{\left(27 n +77\right) a \! \left(n +2\right)}{n +5}+\frac{\left(35+9 n \right) a \! \left(n +3\right)}{n +5}+\frac{\left(n +1\right) \left(n^{3}-6 n^{2}-7 n +10\right)}{2 n +10}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 262\)
\(\displaystyle a \! \left(7\right) = 896\)
\(\displaystyle a \! \left(8\right) = 3033\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(2 n +3\right) a \! \left(n \right)}{n +5}+\frac{2 \left(15 n +28\right) a \! \left(n +1\right)}{n +5}-\frac{\left(27 n +77\right) a \! \left(n +2\right)}{n +5}+\frac{\left(35+9 n \right) a \! \left(n +3\right)}{n +5}+\frac{\left(n +1\right) \left(n^{3}-6 n^{2}-7 n +10\right)}{2 n +10}, \quad n \geq 9\)
This specification was found using the strategy pack "Insertion Point Placements" and has 42 rules.
Found on July 23, 2021.Finding the specification took 10 seconds.
Copy 42 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_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{6} \left(x \right)^{2} F_{20}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{23}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{37}\! \left(x \right) F_{6}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{40}\! \left(x \right) F_{6}\! \left(x \right)\\
\end{align*}\)