Av(1243, 1324, 2143, 2413, 3142)
Generating Function
\(\displaystyle -\frac{2 \left(\left(x^{2}-\frac{3}{2} x +\frac{1}{2}\right) \sqrt{1-4 x}+x^{3}-x^{2}+\frac{3 x}{2}-\frac{1}{2}\right) \left(x -1\right) \left(x -\frac{1}{2}\right)}{x \left(x^{5}+2 x^{4}-9 x^{3}+12 x^{2}-6 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 60, 191, 618, 2036, 6823, 23220, 80113, 279775, 987534, 3518606, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{5}+2 x^{4}-9 x^{3}+12 x^{2}-6 x +1\right) F \left(x
\right)^{2}+\left(x -1\right) \left(2 x -1\right) \left(2 x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} \left(2 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 191\)
\(\displaystyle a \! \left(7\right) = 618\)
\(\displaystyle a \! \left(n +8\right) = \frac{4 \left(2 n +3\right) a \! \left(n \right)}{n +9}+\frac{2 \left(n -18\right) a \! \left(1+n \right)}{n +9}-\frac{3 \left(31 n +60\right) a \! \left(n +2\right)}{n +9}+\frac{\left(771+235 n \right) a \! \left(n +3\right)}{n +9}-\frac{\left(281 n +1242\right) a \! \left(n +4\right)}{n +9}+\frac{\left(1029+185 n \right) a \! \left(n +5\right)}{n +9}-\frac{4 \left(17 n +114\right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 191\)
\(\displaystyle a \! \left(7\right) = 618\)
\(\displaystyle a \! \left(n +8\right) = \frac{4 \left(2 n +3\right) a \! \left(n \right)}{n +9}+\frac{2 \left(n -18\right) a \! \left(1+n \right)}{n +9}-\frac{3 \left(31 n +60\right) a \! \left(n +2\right)}{n +9}+\frac{\left(771+235 n \right) a \! \left(n +3\right)}{n +9}-\frac{\left(281 n +1242\right) a \! \left(n +4\right)}{n +9}+\frac{\left(1029+185 n \right) a \! \left(n +5\right)}{n +9}-\frac{4 \left(17 n +114\right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 23 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 23 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= x\\
F_{8}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
\end{align*}\)