Av(1342, 2341, 2413, 3412)
Generating Function
\(\displaystyle -\frac{\left(3 \sqrt{1-4 x}\, x^{2}-3 \sqrt{1-4 x}\, x -x^{2}+\sqrt{1-4 x}+3 x -1\right) \left(x -1\right) \left(2 x -1\right)}{18 x^{5}-40 x^{4}+36 x^{3}-14 x^{2}+2 x}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 224, 753, 2557, 8782, 30493, 106940, 378423, 1349927, 4850432, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(9 x^{4}-20 x^{3}+18 x^{2}-7 x +1\right) F \left(x
\right)^{2}-\left(2 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+\left(2 x -1\right)^{2} \left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 224\)
\(\displaystyle a \! \left(7\right) = 753\)
\(\displaystyle a \! \left(8\right) = 2557\)
\(\displaystyle a \! \left(n +9\right) = \frac{108 \left(1+2 n \right) a \! \left(n \right)}{n +10}-\frac{6 \left(283+179 n \right) a \! \left(1+n \right)}{n +10}+\frac{3 \left(2088+797 n \right) a \! \left(n +2\right)}{n +10}-\frac{2 \left(5700+1559 n \right) a \! \left(n +3\right)}{n +10}+\frac{2 \left(6150+1307 n \right) a \! \left(n +4\right)}{n +10}-\frac{4 \left(2099+364 n \right) a \! \left(n +5\right)}{n +10}+\frac{\left(3668+537 n \right) a \! \left(n +6\right)}{n +10}-\frac{14 \left(71+9 n \right) a \! \left(n +7\right)}{n +10}+\frac{\left(152+17 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 224\)
\(\displaystyle a \! \left(7\right) = 753\)
\(\displaystyle a \! \left(8\right) = 2557\)
\(\displaystyle a \! \left(n +9\right) = \frac{108 \left(1+2 n \right) a \! \left(n \right)}{n +10}-\frac{6 \left(283+179 n \right) a \! \left(1+n \right)}{n +10}+\frac{3 \left(2088+797 n \right) a \! \left(n +2\right)}{n +10}-\frac{2 \left(5700+1559 n \right) a \! \left(n +3\right)}{n +10}+\frac{2 \left(6150+1307 n \right) a \! \left(n +4\right)}{n +10}-\frac{4 \left(2099+364 n \right) a \! \left(n +5\right)}{n +10}+\frac{\left(3668+537 n \right) a \! \left(n +6\right)}{n +10}-\frac{14 \left(71+9 n \right) a \! \left(n +7\right)}{n +10}+\frac{\left(152+17 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 26 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 26 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)