Av(1243, 2143, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(-x^{4}+2 x^{3}+x^{2}-2 x +1\right) \sqrt{1-4 x}+5 x^{4}-4 x^{3}+x^{2}+2 x -1}{2 x \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 214, 694, 2280, 7625, 25957, 89763, 314573, 1114779, 3987809, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x
\right)^{2}-\left(5 x^{4}-4 x^{3}+x^{2}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}+2 x^{7}-7 x^{6}+14 x^{5}-8 x^{4}-4 x^{3}+9 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 214\)
\(\displaystyle a \! \left(7\right) = 694\)
\(\displaystyle a \! \left(8\right) = 2280\)
\(\displaystyle a \! \left(n +6\right) = \frac{2 \left(-1+2 n \right) a \! \left(n \right)}{n +7}-\frac{\left(3+13 n \right) a \! \left(n +1\right)}{n +7}+\frac{\left(-13+7 n \right) a \! \left(n +2\right)}{n +7}+\frac{\left(53+11 n \right) a \! \left(n +3\right)}{n +7}-\frac{15 \left(n +5\right) a \! \left(n +4\right)}{n +7}+\frac{\left(41+7 n \right) a \! \left(n +5\right)}{n +7}+\frac{6 n}{n +7}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 214\)
\(\displaystyle a \! \left(7\right) = 694\)
\(\displaystyle a \! \left(8\right) = 2280\)
\(\displaystyle a \! \left(n +6\right) = \frac{2 \left(-1+2 n \right) a \! \left(n \right)}{n +7}-\frac{\left(3+13 n \right) a \! \left(n +1\right)}{n +7}+\frac{\left(-13+7 n \right) a \! \left(n +2\right)}{n +7}+\frac{\left(53+11 n \right) a \! \left(n +3\right)}{n +7}-\frac{15 \left(n +5\right) a \! \left(n +4\right)}{n +7}+\frac{\left(41+7 n \right) a \! \left(n +5\right)}{n +7}+\frac{6 n}{n +7}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 21 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 21 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9} \left(x \right)^{3}\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18} \left(x \right)^{2} F_{10}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18} \left(x \right)^{2} F_{10}\! \left(x \right) F_{9}\! \left(x \right)\\
\end{align*}\)