Av(12435, 12453, 14235, 14253, 41235, 41253)
Counting Sequence
1, 1, 2, 6, 24, 114, 596, 3298, 18944, 111778, 673220, 4121434, 25570144, 160415810, 1015899124, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) F \left(x
\right)^{3}+\left(-3 x +1\right) F \left(x
\right)^{2}+2 F \! \left(x \right)-2 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(n +5\right) = \frac{81 n \left(2 n +1\right) a \! \left(n \right)}{\left(n +5\right) \left(n +4\right)}-\frac{9 \left(72 n^{2}+153 n +80\right) a \! \left(n +1\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{3 \left(n +2\right) \left(167 n +300\right) a \! \left(n +2\right)}{2 \left(n +5\right) \left(n +4\right)}-\frac{\left(185 n^{2}+1021 n +1400\right) a \! \left(n +3\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{\left(32 n +105\right) a \! \left(n +4\right)}{2 n +10}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(n +5\right) = \frac{81 n \left(2 n +1\right) a \! \left(n \right)}{\left(n +5\right) \left(n +4\right)}-\frac{9 \left(72 n^{2}+153 n +80\right) a \! \left(n +1\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{3 \left(n +2\right) \left(167 n +300\right) a \! \left(n +2\right)}{2 \left(n +5\right) \left(n +4\right)}-\frac{\left(185 n^{2}+1021 n +1400\right) a \! \left(n +3\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{\left(32 n +105\right) a \! \left(n +4\right)}{2 n +10}, \quad n \geq 5\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 26 rules.
Found on January 22, 2022.Finding the specification took 10 seconds.
Copy 26 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_{9}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x , y\right)\\
F_{5}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{4}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x , y\right) &= \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{13}\! \left(x , y\right) F_{6}\! \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_{17}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
\end{align*}\)