Av(1234, 1243, 1342, 3124)
Generating Function
\(\displaystyle \frac{\left(-2 x^{4}+4 x^{3}-5 x^{2}+3 x -1\right) \sqrt{1-4 x}-4 x^{4}+12 x^{3}-13 x^{2}+7 x -1}{8 x^{5}-26 x^{4}+40 x^{3}-32 x^{2}+14 x -2}\)
Counting Sequence
1, 1, 2, 6, 20, 68, 235, 827, 2956, 10693, 39039, 143586, 531372, 1976777, 7387138, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{5}-13 x^{4}+20 x^{3}-16 x^{2}+7 x -1\right) F \left(x
\right)^{2}+\left(4 x^{4}-12 x^{3}+13 x^{2}-7 x +1\right) F \! \left(x \right)+x \left(x^{3}-x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 235\)
\(\displaystyle a \! \left(7\right) = 827\)
\(\displaystyle a \! \left(8\right) = 2956\)
\(\displaystyle a \! \left(9\right) = 10693\)
\(\displaystyle a \! \left(n +10\right) = -\frac{16 \left(1+2 n \right) a \! \left(n \right)}{n +10}+\frac{4 \left(57+44 n \right) a \! \left(n +1\right)}{n +10}-\frac{2 \left(538+245 n \right) a \! \left(n +2\right)}{n +10}+\frac{2 \left(1377+434 n \right) a \! \left(n +3\right)}{n +10}-\frac{\left(4488+1073 n \right) a \! \left(n +4\right)}{n +10}+\frac{\left(4962+953 n \right) a \! \left(n +5\right)}{n +10}-\frac{\left(3804+611 n \right) a \! \left(n +6\right)}{n +10}+\frac{5 \left(398+55 n \right) a \! \left(n +7\right)}{n +10}-\frac{2 \left(337+41 n \right) a \! \left(n +8\right)}{n +10}+\frac{2 \left(64+7 n \right) a \! \left(n +9\right)}{n +10}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 235\)
\(\displaystyle a \! \left(7\right) = 827\)
\(\displaystyle a \! \left(8\right) = 2956\)
\(\displaystyle a \! \left(9\right) = 10693\)
\(\displaystyle a \! \left(n +10\right) = -\frac{16 \left(1+2 n \right) a \! \left(n \right)}{n +10}+\frac{4 \left(57+44 n \right) a \! \left(n +1\right)}{n +10}-\frac{2 \left(538+245 n \right) a \! \left(n +2\right)}{n +10}+\frac{2 \left(1377+434 n \right) a \! \left(n +3\right)}{n +10}-\frac{\left(4488+1073 n \right) a \! \left(n +4\right)}{n +10}+\frac{\left(4962+953 n \right) a \! \left(n +5\right)}{n +10}-\frac{\left(3804+611 n \right) a \! \left(n +6\right)}{n +10}+\frac{5 \left(398+55 n \right) a \! \left(n +7\right)}{n +10}-\frac{2 \left(337+41 n \right) a \! \left(n +8\right)}{n +10}+\frac{2 \left(64+7 n \right) a \! \left(n +9\right)}{n +10}, \quad n \geq 10\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 26 rules.
Found on July 23, 2021.Finding the specification took 3 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_{22}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{5}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= y x\\
F_{10}\! \left(x , y\right) &= F_{1}\! \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_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{17}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= \frac{y F_{14}\! \left(x , y\right)-F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{21}\! \left(x , y\right) &= \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{22}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{23}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= \frac{y F_{17}\! \left(x , y\right)-F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{25}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\
\end{align*}\)