Av(1243, 1324, 1342, 2143, 2431)
Generating Function
\(\displaystyle \frac{\left(-2 x^{3}+x^{2}+x -1\right) \sqrt{1-4 x}-2 x^{4}+4 x^{3}-3 x^{2}-x +1}{2 x \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 185, 594, 1956, 6585, 22576, 78555, 276688, 984515, 3533442, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(2 x^{4}-4 x^{3}+3 x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{7}+2 x^{5}-7 x^{4}+6 x^{3}+x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 185\)
\(\displaystyle a \! \left(7\right) = 594\)
\(\displaystyle a \! \left(n +5\right) = -\frac{4 \left(-1+2 n \right) a \! \left(n \right)}{6+n}+\frac{2 \left(-2+7 n \right) a \! \left(1+n \right)}{6+n}-\frac{3 \left(n -3\right) a \! \left(n +2\right)}{6+n}-\frac{\left(35+8 n \right) a \! \left(n +3\right)}{6+n}+\frac{\left(29+6 n \right) a \! \left(n +4\right)}{6+n}-\frac{1}{6+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 185\)
\(\displaystyle a \! \left(7\right) = 594\)
\(\displaystyle a \! \left(n +5\right) = -\frac{4 \left(-1+2 n \right) a \! \left(n \right)}{6+n}+\frac{2 \left(-2+7 n \right) a \! \left(1+n \right)}{6+n}-\frac{3 \left(n -3\right) a \! \left(n +2\right)}{6+n}-\frac{\left(35+8 n \right) a \! \left(n +3\right)}{6+n}+\frac{\left(29+6 n \right) a \! \left(n +4\right)}{6+n}-\frac{1}{6+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 24 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 24 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{0}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)\\
\end{align*}\)