Av(1324, 1432, 2341, 2413, 3214)
Generating Function
\(\displaystyle \frac{2 x^{7}-4 x^{6}+x^{5}+4 x^{4}+3 x^{2}-3 x +1}{\left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 50, 118, 269, 598, 1302, 2803, 6002, 12806, 27261, 57966, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-2 x^{7}+4 x^{6}-x^{5}-4 x^{4}-3 x^{2}+3 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 118\)
\(\displaystyle a \! \left(7\right) = 269\)
\(\displaystyle a \! \left(n +4\right) = 2 n^{2}+2 a \! \left(n \right)+2 a \! \left(n +1\right)+a \! \left(n +2\right)+a \! \left(n +3\right)+8 n +9, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 118\)
\(\displaystyle a \! \left(7\right) = 269\)
\(\displaystyle a \! \left(n +4\right) = 2 n^{2}+2 a \! \left(n \right)+2 a \! \left(n +1\right)+a \! \left(n +2\right)+a \! \left(n +3\right)+8 n +9, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle -\frac{151}{125}+\frac{3028 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}+2 Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{6625}+\frac{13858 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}+2 Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{19875}+\frac{1561 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}+2 Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{3975}+\frac{6647 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}+2 Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{19875}-\frac{28 n}{25}-\frac{2 n^{2}}{5}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 49 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 49 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_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{33}\! \left(x \right)\\
\end{align*}\)