Av(1234, 1243, 2431, 3214, 4213)
Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(3 x^{10}+5 x^{9}-x^{8}-12 x^{7}-14 x^{6}+x^{5}+7 x^{4}+x^{3}+x^{2}-2 x +1\right)}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 47, 89, 166, 317, 602, 1138, 2143, 4022, 7528, 14057, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+\left(x +1\right) \left(3 x^{10}+5 x^{9}-x^{8}-12 x^{7}-14 x^{6}+x^{5}+7 x^{4}+x^{3}+x^{2}-2 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 47\)
\(\displaystyle a \! \left(6\right) = 89\)
\(\displaystyle a \! \left(7\right) = 166\)
\(\displaystyle a \! \left(8\right) = 317\)
\(\displaystyle a \! \left(9\right) = 602\)
\(\displaystyle a \! \left(10\right) = 1138\)
\(\displaystyle a \! \left(11\right) = 2143\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right), \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 47\)
\(\displaystyle a \! \left(6\right) = 89\)
\(\displaystyle a \! \left(7\right) = 166\)
\(\displaystyle a \! \left(8\right) = 317\)
\(\displaystyle a \! \left(9\right) = 602\)
\(\displaystyle a \! \left(10\right) = 1138\)
\(\displaystyle a \! \left(11\right) = 2143\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right), \quad n \geq 12\)
Explicit Closed Form
\(\displaystyle -\frac{273 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{3-n}\right)}{55}-\frac{772 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{2-n}\right)}{55}-\frac{762 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{55}-\frac{191 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{55}+\frac{569 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{55}-\left(\left\{\begin{array}{cc}-3 & n =0 \\ 3 & n =1\text{ or } n =2 \\ 6 & n =3 \\ 3 & n =4 \\ -2 & n =5 \\ -3 & n =6 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 75 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 75 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_{24}\! \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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \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_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \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_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{33}\! \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_{34}\! \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_{65}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{35}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{71}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
\end{align*}\)