Av(1234, 1342, 3124)
Generating Function
\(\displaystyle \frac{-4 \left(x^{2}-x +\frac{1}{2}\right)^{2} x \sqrt{1-4 x}+12 x^{5}-52 x^{4}+76 x^{3}-52 x^{2}+17 x -2}{8 x^{6}+16 x^{5}-76 x^{4}+98 x^{3}-60 x^{2}+18 x -2}\)
Counting Sequence
1, 1, 2, 6, 21, 76, 278, 1029, 3859, 14642, 56067, 216174, 837832, 3260369, 12728853, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{6}+8 x^{5}-38 x^{4}+49 x^{3}-30 x^{2}+9 x -1\right) F \left(x
\right)^{2}+\left(-12 x^{5}+52 x^{4}-76 x^{3}+52 x^{2}-17 x +2\right) F \! \left(x \right)+4 x^{5}-16 x^{4}+28 x^{3}-22 x^{2}+8 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 278\)
\(\displaystyle a \! \left(7\right) = 1029\)
\(\displaystyle a \! \left(8\right) = 3859\)
\(\displaystyle a \! \left(n +9\right) = -\frac{16 \left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{3 \left(36+5 n \right) a \! \left(8+n \right)}{8+n}-\frac{24 \left(n -4\right) a \! \left(n +1\right)}{8+n}+\frac{24 \left(11+15 n \right) a \! \left(n +2\right)}{8+n}-\frac{12 \left(139+68 n \right) a \! \left(n +3\right)}{8+n}+\frac{6 \left(520+161 n \right) a \! \left(n +4\right)}{8+n}-\frac{8 \left(383+88 n \right) a \! \left(n +5\right)}{8+n}+\frac{3 \left(588+109 n \right) a \! \left(n +6\right)}{8+n}-\frac{2 \left(298+47 n \right) a \! \left(n +7\right)}{8+n}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 278\)
\(\displaystyle a \! \left(7\right) = 1029\)
\(\displaystyle a \! \left(8\right) = 3859\)
\(\displaystyle a \! \left(n +9\right) = -\frac{16 \left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{3 \left(36+5 n \right) a \! \left(8+n \right)}{8+n}-\frac{24 \left(n -4\right) a \! \left(n +1\right)}{8+n}+\frac{24 \left(11+15 n \right) a \! \left(n +2\right)}{8+n}-\frac{12 \left(139+68 n \right) a \! \left(n +3\right)}{8+n}+\frac{6 \left(520+161 n \right) a \! \left(n +4\right)}{8+n}-\frac{8 \left(383+88 n \right) a \! \left(n +5\right)}{8+n}+\frac{3 \left(588+109 n \right) a \! \left(n +6\right)}{8+n}-\frac{2 \left(298+47 n \right) a \! \left(n +7\right)}{8+n}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 29 rules.
Found on July 23, 2021.Finding the specification took 58 seconds.
Copy 29 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_{17}\! \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_{17}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= y x\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{16}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{17}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{25}\! \left(x \right) F_{26}\! \left(x , y\right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x , 1\right)\\
F_{21}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= -\frac{-y F_{21}\! \left(x , y\right)+F_{21}\! \left(x , 1\right)}{-1+y}\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x , 1\right)\\
F_{26}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
\end{align*}\)