Av(1324, 1342, 1423, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(x^{3}+2 x^{2}-4 x +2\right) \sqrt{1-4 x}+2 x^{4}+x^{3}-6 x^{2}+6 x -2}{2 \left(x -1\right) x}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 188, 616, 2066, 7060, 24493, 86035, 305369, 1093489, 3945579, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{2} x F \left(x
\right)^{2}-\left(x -1\right) \left(2 x^{4}+x^{3}-6 x^{2}+6 x -2\right) F \! \left(x \right)+x^{7}+2 x^{6}-2 x^{5}-2 x^{4}-x^{3}+8 x^{2}-7 x +2 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 188\)
\(\displaystyle a \! \left(7\right) = 616\)
\(\displaystyle a \! \left(n +5\right) = -\frac{\left(-3+2 n \right) a \! \left(n \right)}{n +6}-\frac{\left(10+3 n \right) a \! \left(1+n \right)}{2 \left(n +6\right)}+\frac{\left(25 n +48\right) a \! \left(n +2\right)}{12+2 n}-\frac{3 \left(5 n +16\right) a \! \left(n +3\right)}{n +6}+\frac{\left(7 n +32\right) a \! \left(n +4\right)}{n +6}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 188\)
\(\displaystyle a \! \left(7\right) = 616\)
\(\displaystyle a \! \left(n +5\right) = -\frac{\left(-3+2 n \right) a \! \left(n \right)}{n +6}-\frac{\left(10+3 n \right) a \! \left(1+n \right)}{2 \left(n +6\right)}+\frac{\left(25 n +48\right) a \! \left(n +2\right)}{12+2 n}-\frac{3 \left(5 n +16\right) a \! \left(n +3\right)}{n +6}+\frac{\left(7 n +32\right) a \! \left(n +4\right)}{n +6}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 19 rules.
Found on July 23, 2021.Finding the specification took 1 seconds.
Copy 19 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{5} \left(x \right)^{3}\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{6}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
\end{align*}\)