Av(1243, 1432, 2413, 2431, 4132)
Generating Function
\(\displaystyle -\frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{4}-3 x^{3}+6 x^{2}-4 x +1\right)}{2 x \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 175, 538, 1709, 5612, 18944, 65315, 228823, 811608, 2907160, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{8} F \left(x
\right)^{2}-\left(x^{4}-3 x^{3}+6 x^{2}-4 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+\left(x^{4}-3 x^{3}+6 x^{2}-4 x +1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 175\)
\(\displaystyle a \! \left(7\right) = 538\)
\(\displaystyle a \! \left(8\right) = 1709\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{\left(22+17 n \right) a \! \left(n +1\right)}{n +7}-\frac{\left(40 n +113\right) a \! \left(n +2\right)}{n +7}+\frac{\left(179+49 n \right) a \! \left(n +3\right)}{n +7}-\frac{10 \left(3 n +14\right) a \! \left(n +4\right)}{n +7}+\frac{\left(52+9 n \right) a \! \left(n +5\right)}{n +7}+\frac{\left(n +1\right) \left(n +2\right)}{2 n +14}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 175\)
\(\displaystyle a \! \left(7\right) = 538\)
\(\displaystyle a \! \left(8\right) = 1709\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{\left(22+17 n \right) a \! \left(n +1\right)}{n +7}-\frac{\left(40 n +113\right) a \! \left(n +2\right)}{n +7}+\frac{\left(179+49 n \right) a \! \left(n +3\right)}{n +7}-\frac{10 \left(3 n +14\right) a \! \left(n +4\right)}{n +7}+\frac{\left(52+9 n \right) a \! \left(n +5\right)}{n +7}+\frac{\left(n +1\right) \left(n +2\right)}{2 n +14}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 24 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 24 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{6}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right) F_{22}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
\end{align*}\)