Av(1234, 1243, 1324, 1342, 1423, 2134, 2314, 2341, 3124, 3412)
Generating Function
\(\displaystyle -\frac{2 x^{7}-7 x^{6}+9 x^{5}-x^{4}-8 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 14, 34, 81, 186, 412, 886, 1863, 3854, 7882, 15994, 32285, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+2 x^{7}-7 x^{6}+9 x^{5}-x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 14\)
\(\displaystyle a \! \left(5\right) = 34\)
\(\displaystyle a \! \left(6\right) = 81\)
\(\displaystyle a \! \left(7\right) = 186\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(-2+n \right) \left(n^{2}-n +6\right)}{6}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 14\)
\(\displaystyle a \! \left(5\right) = 34\)
\(\displaystyle a \! \left(6\right) = 81\)
\(\displaystyle a \! \left(7\right) = 186\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(-2+n \right) \left(n^{2}-n +6\right)}{6}, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 2^{n +1}-\frac{11 n}{6}-\frac{n^{3}}{6} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 31 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 31 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_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \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_{16}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14} \left(x \right)^{2} F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)