Av(1342, 2314, 2341, 2413, 3142)
Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{3} \sqrt{-4 x +1}+3 x^{3}-3 x^{2}+3 x -1}{2 \left(x^{3}-2 x^{2}+3 x -1\right) x}\)
Counting Sequence
1, 1, 2, 6, 19, 60, 191, 617, 2026, 6762, 22922, 78822, 274564, 967396, 3442872, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-2 x^{2}+3 x -1\right)^{2} x F \left(x
\right)^{2}-\left(3 x^{3}-3 x^{2}+3 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +6}-\frac{\left(13 n +24\right) a \! \left(1+n \right)}{n +6}+\frac{\left(52+23 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(21 n +74\right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(4 n +19\right) a \! \left(n +4\right)}{n +6}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +6}-\frac{\left(13 n +24\right) a \! \left(1+n \right)}{n +6}+\frac{\left(52+23 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(21 n +74\right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(4 n +19\right) a \! \left(n +4\right)}{n +6}, \quad n \geq 5\)
This specification was found using the strategy pack "Point Placements" and has 28 rules.
Found on July 23, 2021.Finding the specification took 1 seconds.
Copy 28 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{25}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{25} \left(x \right)^{2} F_{8}\! \left(x \right)\\
\end{align*}\)