Av(1243, 1342, 1423, 2134, 2341)
Generating Function
\(\displaystyle \frac{-\left(x -1\right) \left(x^{3}-x +1\right) \sqrt{1-4 x}-2 x^{5}+3 x^{4}+x^{3}-x^{2}+2 x -1}{4 x^{2}-2 x}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 181, 578, 1888, 6297, 21393, 73843, 258354, 914304, 3267250, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} F \left(x
\right)^{2}+\left(x +1\right) \left(x^{3}-2 x^{2}+x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{9}-2 x^{8}-x^{7}+2 x^{6}+3 x^{5}-3 x^{4}-x^{3}+5 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 181\)
\(\displaystyle a \! \left(7\right) = 578\)
\(\displaystyle a \! \left(8\right) = 1888\)
\(\displaystyle a \! \left(9\right) = 6297\)
\(\displaystyle a \! \left(n +6\right) = \frac{4 \left(-5+2 n \right) a \! \left(n \right)}{n +7}-\frac{2 \left(-8+7 n \right) a \! \left(1+n \right)}{n +7}-\frac{\left(13+n \right) a \! \left(n +2\right)}{n +7}+\frac{\left(67+21 n \right) a \! \left(n +3\right)}{n +7}-\frac{\left(95+21 n \right) a \! \left(n +4\right)}{n +7}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{n +7}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 181\)
\(\displaystyle a \! \left(7\right) = 578\)
\(\displaystyle a \! \left(8\right) = 1888\)
\(\displaystyle a \! \left(9\right) = 6297\)
\(\displaystyle a \! \left(n +6\right) = \frac{4 \left(-5+2 n \right) a \! \left(n \right)}{n +7}-\frac{2 \left(-8+7 n \right) a \! \left(1+n \right)}{n +7}-\frac{\left(13+n \right) a \! \left(n +2\right)}{n +7}+\frac{\left(67+21 n \right) a \! \left(n +3\right)}{n +7}-\frac{\left(95+21 n \right) a \! \left(n +4\right)}{n +7}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{n +7}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 41 rules.
Found on January 20, 2022.Finding the specification took 5 seconds.
Copy 41 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_{30}\! \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_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x , 1\right)\\
F_{33}\! \left(x , y\right) &= \frac{y F_{34}\! \left(x , y\right)-F_{34}\! \left(x , 1\right)}{-1+y}\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{35}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= y x\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{35}\! \left(x , y\right) F_{8}\! \left(x \right)\\
\end{align*}\)