Av(1234, 1432, 2341, 2413, 3142)
Generating Function
\(\displaystyle -\frac{x^{8}+2 x^{7}+2 x^{6}+2 x^{5}-5 x^{4}-2 x^{2}+3 x -1}{\left(x^{3}+2 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 50, 125, 311, 765, 1860, 4483, 10728, 25515, 60363, 142151, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+2 x -1\right)^{2} F \! \left(x \right)+x^{8}+2 x^{7}+2 x^{6}+2 x^{5}-5 x^{4}-2 x^{2}+3 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 125\)
\(\displaystyle a \! \left(7\right) = 311\)
\(\displaystyle a \! \left(8\right) = 765\)
\(\displaystyle a \! \left(n \right) = -4 a \! \left(n +2\right)+2 a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right)-a \! \left(n +6\right), \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 125\)
\(\displaystyle a \! \left(7\right) = 311\)
\(\displaystyle a \! \left(8\right) = 765\)
\(\displaystyle a \! \left(n \right) = -4 a \! \left(n +2\right)+2 a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right)-a \! \left(n +6\right), \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \frac{\left(236 n +236\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+2 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -2}\right)}{3481}+\frac{\left(-59 n -219\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+2 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{3481}+\frac{3361 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+2 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{3481}+\frac{\left(-531 n -1329\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+2 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{3481}-\left(\left\{\begin{array}{cc}-2 & n =0 \\ 2 & n =1 \\ 1 & n =2 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 64 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 64 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_{23}\! \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_{14}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \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_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{7}\! \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_{11}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{54}\! \left(x \right)\\
\end{align*}\)