Av(1243, 1342, 1423, 2134)
Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{2} \sqrt{-4 x +1}+2 x^{5}+2 x^{4}+4 x^{3}+3 x^{2}-4 x +1}{2 x^{3} \left(x^{2}+3\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 228, 788, 2758, 9756, 34826, 125302, 453942, 1654521, 6062826, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{3} \left(x^{2}+3\right) F \left(x
\right)^{2}+\left(-2 x^{5}-2 x^{4}-4 x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)+x^{5}+2 x^{4}+2 x^{3}+x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(5+2 n \right) a \! \left(n \right)}{3 \left(7+n \right)}+\frac{\left(26+5 n \right) a \! \left(3+n \right)}{7+n}+\frac{\left(26+5 n \right) a \! \left(n +1\right)}{21+3 n}-\frac{\left(37+13 n \right) a \! \left(n +2\right)}{3 \left(7+n \right)}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(5+2 n \right) a \! \left(n \right)}{3 \left(7+n \right)}+\frac{\left(26+5 n \right) a \! \left(3+n \right)}{7+n}+\frac{\left(26+5 n \right) a \! \left(n +1\right)}{21+3 n}-\frac{\left(37+13 n \right) a \! \left(n +2\right)}{3 \left(7+n \right)}, \quad n \geq 5\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 25 rules.
Found on July 23, 2021.Finding the specification took 6 seconds.
Copy 25 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{11}\! \left(x \right) F_{3}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{17}\! \left(x , y\right)+F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{16}\! \left(x , y\right) &= y x\\
F_{17}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= \frac{y F_{14}\! \left(x , y\right)-F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x , 1\right)\\
F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y\right)+F_{23}\! \left(x , y\right)+F_{4}\! \left(x \right)\\
F_{22}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= \frac{y F_{21}\! \left(x , y\right)-F_{21}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)