Av(1243, 1324, 1432, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(-2 x^{2}+3 x -1\right) \sqrt{1-4 x}-2 x^{3}+2 x^{2}-3 x +1}{2 \left(x^{2}-3 x +1\right) x}\)
Counting Sequence
1, 1, 2, 6, 19, 61, 198, 650, 2159, 7257, 24682, 84911, 295297, 1037416, 3678715, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}-3 x +1\right)^{2} F \left(x
\right)^{2}+\left(x^{2}-3 x +1\right) \left(2 x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{5}+2 x^{4}-9 x^{3}+12 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(n +5\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +6}-\frac{2 \left(23+19 n \right) a \! \left(1+n \right)}{n +6}+\frac{\left(136+57 n \right) a \! \left(n +2\right)}{n +6}-\frac{2 \left(65+18 n \right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(24+5 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(n +5\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +6}-\frac{2 \left(23+19 n \right) a \! \left(1+n \right)}{n +6}+\frac{\left(136+57 n \right) a \! \left(n +2\right)}{n +6}-\frac{2 \left(65+18 n \right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(24+5 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 6\)
This specification was found using the strategy pack "Point Placements" and has 26 rules.
Found on July 23, 2021.Finding the specification took 7 seconds.
Copy 26 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{5}\! \left(x \right)\\
\end{align*}\)