Av(1234, 1342, 2314, 2413, 3412)
Generating Function
\(\displaystyle -\frac{\left(x^{2}-x +1\right) \left(3 x^{4}-13 x^{3}+14 x^{2}-6 x +1\right)}{\left(2 x -1\right) \left(x -1\right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 19, 55, 143, 340, 756, 1603, 3292, 6626, 13185, 26095, 51565, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x -1\right)^{6} F \! \left(x \right)+\left(x^{2}-x +1\right) \left(3 x^{4}-13 x^{3}+14 x^{2}-6 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 143\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-9 n^{3}-14 n^{2}-4 n -120\right)}{120}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 143\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-9 n^{3}-14 n^{2}-4 n -120\right)}{120}, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{143 n}{60}-2+\frac{n^{5}}{120}-\frac{n^{4}}{24}-\frac{n^{3}}{8}-\frac{11 n^{2}}{24}+3 \,2^{n}\)
This specification was found using the strategy pack "Point Placements" and has 42 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 42 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_{21}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \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_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 0\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{39}\! \left(x \right)\\
\end{align*}\)