Av(1243, 1324, 1342, 1423, 2143, 2413)
Generating Function
\(\displaystyle \frac{\left(x^{2}+x -1+\left(x^{2}-x +1\right) \sqrt{1-4 x}\right) \left(x -1\right)}{2 x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 18, 55, 174, 567, 1891, 6426, 22177, 77530, 274013, 977499, 3515206, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) F \left(x
\right)^{2}-\left(x -1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 174\)
\(\displaystyle a \! \left(7\right) = 567\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{9+n}+\frac{\left(17 n +43\right) a \! \left(1+n \right)}{9+n}-\frac{4 \left(11 n +35\right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(37 n +147\right) a \! \left(n +3\right)}{9+n}-\frac{4 \left(21 n +103\right) a \! \left(n +4\right)}{9+n}+\frac{\left(65 n +387\right) a \! \left(n +5\right)}{9+n}-\frac{2 \left(16 n +111\right) a \! \left(n +6\right)}{9+n}+\frac{\left(9 n +71\right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 174\)
\(\displaystyle a \! \left(7\right) = 567\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{9+n}+\frac{\left(17 n +43\right) a \! \left(1+n \right)}{9+n}-\frac{4 \left(11 n +35\right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(37 n +147\right) a \! \left(n +3\right)}{9+n}-\frac{4 \left(21 n +103\right) a \! \left(n +4\right)}{9+n}+\frac{\left(65 n +387\right) a \! \left(n +5\right)}{9+n}-\frac{2 \left(16 n +111\right) a \! \left(n +6\right)}{9+n}+\frac{\left(9 n +71\right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 15 rules.
Found on July 23, 2021.Finding the specification took 1 seconds.
Copy 15 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
\end{align*}\)