Av(1342, 13254)
Counting Sequence
1, 1, 2, 6, 23, 102, 495, 2549, 13682, 75714, 428882, 2474573, 14492346, 85926361, 514763279, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-x +1\right) F \left(x
\right)^{3}+\left(x -3\right) F \left(x
\right)^{2}+3 F \! \left(x \right)-1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = \frac{24 \left(3 n +4\right) \left(3 n +2\right) a \! \left(n \right)}{\left(2 n +7\right) \left(n +3\right)}-\frac{\left(469 n^{2}+953 n +378\right) a \! \left(n +1\right)}{2 \left(2 n +7\right) \left(n +3\right)}+\frac{3 \left(155 n^{2}+309 n +112\right) a \! \left(n +2\right)}{2 \left(2 n +7\right) \left(n +3\right)}-\frac{3 \left(11 n^{2}+21 n -16\right) a \! \left(n +3\right)}{2 \left(2 n +7\right) \left(n +3\right)}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = \frac{24 \left(3 n +4\right) \left(3 n +2\right) a \! \left(n \right)}{\left(2 n +7\right) \left(n +3\right)}-\frac{\left(469 n^{2}+953 n +378\right) a \! \left(n +1\right)}{2 \left(2 n +7\right) \left(n +3\right)}+\frac{3 \left(155 n^{2}+309 n +112\right) a \! \left(n +2\right)}{2 \left(2 n +7\right) \left(n +3\right)}-\frac{3 \left(11 n^{2}+21 n -16\right) a \! \left(n +3\right)}{2 \left(2 n +7\right) \left(n +3\right)}, \quad n \geq 4\)
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Req Corrob Expand Verified" and has 22 rules.
Found on January 22, 2022.Finding the specification took 62 seconds.
Copy 22 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{19}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)^{2} F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= y x\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{19}\! \left(x \right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x , y\right)\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x , y\right) &= F_{19}\! \left(x \right) F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Expand Verified" and has 24 rules.
Found on January 22, 2022.Finding the specification took 120 seconds.
Copy 24 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)^{2} F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= y x\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{16}\! \left(x \right) F_{8}\! \left(x , y\right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{7}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= -\frac{y \left(F_{23}\! \left(x , 1\right)-F_{23}\! \left(x , y\right)\right)}{-1+y}\\
F_{8}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{4}\! \left(x \right)\\
\end{align*}\)