Av(1243, 1324, 1342, 1423, 1432, 2143, 2413, 2431, 4132)
Generating Function
\(\displaystyle -\frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{2}-x +1\right) \left(x +1\right)}{2 x}\)
Counting Sequence
1, 1, 2, 6, 15, 44, 137, 443, 1472, 4994, 17225, 60216, 212874, 759696, 2733226, ...
Implicit Equation for the Generating Function
\(\displaystyle x F \left(x
\right)^{2}-\left(x +1\right) \left(x^{2}-x +1\right) F \! \left(x \right)+\left(x +1\right)^{2} \left(x^{2}-x +1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 15\)
\(\displaystyle a \! \left(5\right) = 44\)
\(\displaystyle a \! \left(6\right) = 137\)
\(\displaystyle a \! \left(1+n \right) = \frac{2 \left(-5+2 n \right) a \! \left(n \right)}{n -1}+\frac{2 \left(7+2 n \right) a \! \left(n +3\right)}{n -1}-\frac{\left(5+n \right) a \! \left(n +4\right)}{n -1}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 15\)
\(\displaystyle a \! \left(5\right) = 44\)
\(\displaystyle a \! \left(6\right) = 137\)
\(\displaystyle a \! \left(1+n \right) = \frac{2 \left(-5+2 n \right) a \! \left(n \right)}{n -1}+\frac{2 \left(7+2 n \right) a \! \left(n +3\right)}{n -1}-\frac{\left(5+n \right) a \! \left(n +4\right)}{n -1}, \quad n \geq 7\)
This specification was found using the strategy pack "Point Placements" and has 13 rules.
Found on July 23, 2021.Finding the specification took 1 seconds.
Copy 13 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{5} \left(x \right)^{2}\\
\end{align*}\)