Av(1243, 1342, 2134)
Generating Function
\(\displaystyle \frac{-\left(x^{2}-x +1\right) \left(x -1\right)^{2} \sqrt{1-4 x}+x^{4}+x^{3}-4 x^{2}+5 x -1}{2 x^{5}-8 x^{4}+18 x^{3}-20 x^{2}+12 x -2}\)
Counting Sequence
1, 1, 2, 6, 21, 76, 278, 1026, 3818, 14308, 53932, 204273, 776859, 2964716, 11348261, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-4 x^{4}+9 x^{3}-10 x^{2}+6 x -1\right) F \left(x
\right)^{2}+\left(-x^{4}-x^{3}+4 x^{2}-5 x +1\right) F \! \left(x \right)+x^{2} \left(x^{2}-2 x +2\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 278\)
\(\displaystyle a \! \left(7\right) = 1026\)
\(\displaystyle a \! \left(8\right) = 3818\)
\(\displaystyle a \! \left(n +9\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +9}-\frac{\left(34+25 n \right) a \! \left(n +1\right)}{n +9}+\frac{\left(171+82 n \right) a \! \left(n +2\right)}{n +9}-\frac{\left(493+167 n \right) a \! \left(n +3\right)}{n +9}+\frac{\left(894+229 n \right) a \! \left(n +4\right)}{n +9}-\frac{9 \left(119+24 n \right) a \! \left(n +5\right)}{n +9}+\frac{3 \left(277+46 n \right) a \! \left(n +6\right)}{n +9}-\frac{\left(397+56 n \right) a \! \left(n +7\right)}{n +9}+\frac{\left(97+12 n \right) a \! \left(n +8\right)}{n +9}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 278\)
\(\displaystyle a \! \left(7\right) = 1026\)
\(\displaystyle a \! \left(8\right) = 3818\)
\(\displaystyle a \! \left(n +9\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +9}-\frac{\left(34+25 n \right) a \! \left(n +1\right)}{n +9}+\frac{\left(171+82 n \right) a \! \left(n +2\right)}{n +9}-\frac{\left(493+167 n \right) a \! \left(n +3\right)}{n +9}+\frac{\left(894+229 n \right) a \! \left(n +4\right)}{n +9}-\frac{9 \left(119+24 n \right) a \! \left(n +5\right)}{n +9}+\frac{3 \left(277+46 n \right) a \! \left(n +6\right)}{n +9}-\frac{\left(397+56 n \right) a \! \left(n +7\right)}{n +9}+\frac{\left(97+12 n \right) a \! \left(n +8\right)}{n +9}, \quad n \geq 9\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 28 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 28 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{11}\! \left(x \right) F_{3}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\
F_{13}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{17}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{22}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{16}\! \left(x , y\right) &= y x\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{19}\! \left(x , y\right) F_{3}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\
F_{24}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{25}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{24}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= -\frac{-y F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)