Av(1243, 1342, 2134, 2314, 2341, 4123)
Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{3} \sqrt{-4 x +1}-2 x^{8}+2 x^{7}+2 x^{6}-2 x^{5}-2 x^{4}+x^{3}-3 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 18, 50, 144, 446, 1453, 4892, 16834, 58833, 208069, 742968, 2674520, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x
\right)^{2}+\left(2 x^{8}-2 x^{7}-2 x^{6}+2 x^{5}+2 x^{4}-x^{3}+3 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{15}-2 x^{14}-x^{13}+4 x^{12}+x^{11}-5 x^{10}+3 x^{9}-3 x^{8}+x^{7}+5 x^{6}-7 x^{5}+13 x^{4}-19 x^{3}+15 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 144\)
\(\displaystyle a \! \left(7\right) = 446\)
\(\displaystyle a \! \left(8\right) = 1453\)
\(\displaystyle a \! \left(9\right) = 4892\)
\(\displaystyle a \! \left(n +1\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{3 n^{3}-11 n^{2}+16 n +4}{2 \left(n +2\right)}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 144\)
\(\displaystyle a \! \left(7\right) = 446\)
\(\displaystyle a \! \left(8\right) = 1453\)
\(\displaystyle a \! \left(9\right) = 4892\)
\(\displaystyle a \! \left(n +1\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{3 n^{3}-11 n^{2}+16 n +4}{2 \left(n +2\right)}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 35 rules.
Found on July 23, 2021.Finding the specification took 12 seconds.
Copy 35 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_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{19}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \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) &= y x\\
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_{19}\! \left(x \right)\\
F_{18}\! \left(x , y\right) &= -\frac{y \left(F_{12}\! \left(x , 1\right)-F_{12}\! \left(x , y\right)\right)}{-1+y}\\
F_{19}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{19}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27} \left(x \right)^{3}\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{19}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{19}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{19}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\
\end{align*}\)