Av(1243, 1342, 1423, 2413, 3142)
Generating Function
\(\displaystyle \frac{-x^{3}+x^{2}-2 x +1-\sqrt{x^{6}-6 x^{5}+21 x^{4}-30 x^{3}+22 x^{2}-8 x +1}}{2 x \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 61, 201, 682, 2374, 8436, 30478, 111614, 413384, 1545748, 5827482, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{2} F \left(x
\right)^{2}+\left(x^{3}-x^{2}+2 x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 201\)
\(\displaystyle a \! \left(n +7\right) = \frac{n a \! \left(n \right)}{n +8}-\frac{\left(8+7 n \right) a \! \left(1+n \right)}{n +8}+\frac{3 \left(22+9 n \right) a \! \left(n +2\right)}{n +8}-\frac{3 \left(59+17 n \right) a \! \left(n +3\right)}{n +8}+\frac{\left(237+52 n \right) a \! \left(n +4\right)}{n +8}-\frac{10 \left(17+3 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(61+9 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 201\)
\(\displaystyle a \! \left(n +7\right) = \frac{n a \! \left(n \right)}{n +8}-\frac{\left(8+7 n \right) a \! \left(1+n \right)}{n +8}+\frac{3 \left(22+9 n \right) a \! \left(n +2\right)}{n +8}-\frac{3 \left(59+17 n \right) a \! \left(n +3\right)}{n +8}+\frac{\left(237+52 n \right) a \! \left(n +4\right)}{n +8}-\frac{10 \left(17+3 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(61+9 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7\)
This specification was found using the strategy pack "Point Placements" and has 13 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 13 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
\end{align*}\)