Av(1243, 1324, 2413, 4132)
Generating Function
\(\displaystyle \frac{\left(-2 x^{5}+12 x^{4}-19 x^{3}+15 x^{2}-6 x +1\right) \sqrt{1-4 x}+6 x^{5}-18 x^{4}+21 x^{3}-15 x^{2}+6 x -1}{2 x \left(2 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 213, 681, 2188, 7133, 23700, 80297, 276940, 969839, 3439706, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{8} F \left(x
\right)^{2}-\left(2 x -1\right) \left(6 x^{5}-18 x^{4}+21 x^{3}-15 x^{2}+6 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{10}-40 x^{9}+178 x^{8}-427 x^{7}+640 x^{6}-641 x^{5}+442 x^{4}-209 x^{3}+65 x^{2}-12 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 213\)
\(\displaystyle a \! \left(7\right) = 681\)
\(\displaystyle a \! \left(8\right) = 2188\)
\(\displaystyle a \! \left(9\right) = 7133\)
\(\displaystyle a \! \left(10\right) = 23700\)
\(\displaystyle a \! \left(n +8\right) = -\frac{8 \left(1+2 n \right) a \! \left(n \right)}{n +9}+\frac{4 \left(53+31 n \right) a \! \left(n +1\right)}{n +9}-\frac{2 \left(416+167 n \right) a \! \left(n +2\right)}{n +9}+\frac{8 \left(203+59 n \right) a \! \left(n +3\right)}{n +9}-\frac{\left(1812+403 n \right) a \! \left(n +4\right)}{n +9}+\frac{3 \left(403+72 n \right) a \! \left(n +5\right)}{n +9}-\frac{\left(477+71 n \right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}+\frac{2 n^{2}-4 n -7}{n +9}, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 213\)
\(\displaystyle a \! \left(7\right) = 681\)
\(\displaystyle a \! \left(8\right) = 2188\)
\(\displaystyle a \! \left(9\right) = 7133\)
\(\displaystyle a \! \left(10\right) = 23700\)
\(\displaystyle a \! \left(n +8\right) = -\frac{8 \left(1+2 n \right) a \! \left(n \right)}{n +9}+\frac{4 \left(53+31 n \right) a \! \left(n +1\right)}{n +9}-\frac{2 \left(416+167 n \right) a \! \left(n +2\right)}{n +9}+\frac{8 \left(203+59 n \right) a \! \left(n +3\right)}{n +9}-\frac{\left(1812+403 n \right) a \! \left(n +4\right)}{n +9}+\frac{3 \left(403+72 n \right) a \! \left(n +5\right)}{n +9}-\frac{\left(477+71 n \right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}+\frac{2 n^{2}-4 n -7}{n +9}, \quad n \geq 11\)
This specification was found using the strategy pack "Point Placements" and has 32 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 32 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_{14}\! \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_{14}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
\end{align*}\)