Av(1342, 1423, 2314)
Generating Function
\(\displaystyle -\frac{\left(-3 x +1+\left(x -1\right) \sqrt{-4 x +1}\right) \left(x^{2}-3 x +1\right)}{2 x^{3} \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 76, 276, 1002, 3641, 13261, 48451, 177651, 653753, 2414426, 8947576, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{3} \left(x -1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(3 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+\left(x^{2}-3 x +1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +3\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{6+n}-\frac{\left(32+13 n \right) a \! \left(n +1\right)}{6+n}+\frac{\left(30+7 n \right) a \! \left(n +2\right)}{6+n}+\frac{3 n +6}{6+n}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +3\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{6+n}-\frac{\left(32+13 n \right) a \! \left(n +1\right)}{6+n}+\frac{\left(30+7 n \right) a \! \left(n +2\right)}{6+n}+\frac{3 n +6}{6+n}, \quad n \geq 5\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 28 rules.
Found on July 23, 2021.Finding the specification took 10 seconds.
Copy 28 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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \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_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{13}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{18}\! \left(x \right) F_{21}\! \left(x \right) F_{24}\! \left(x \right) F_{26}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{21}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{21}\! \left(x \right) F_{24}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{24}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{26}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
\end{align*}\)