Av(1234, 1243, 2314, 2431, 3214)
Generating Function
\(\displaystyle -\frac{\left(x +1\right) \left(2 x^{9}+3 x^{8}+2 x^{7}-6 x^{6}-6 x^{5}-7 x^{4}+x^{3}-x^{2}+2 x -1\right)}{\left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 52, 128, 303, 703, 1605, 3610, 8036, 17750, 38953, 85029, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+\left(x +1\right) \left(2 x^{9}+3 x^{8}+2 x^{7}-6 x^{6}-6 x^{5}-7 x^{4}+x^{3}-x^{2}+2 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 128\)
\(\displaystyle a \! \left(7\right) = 303\)
\(\displaystyle a \! \left(8\right) = 703\)
\(\displaystyle a \! \left(9\right) = 1605\)
\(\displaystyle a \! \left(10\right) = 3610\)
\(\displaystyle a \! \left(n +7\right) = -2 a \! \left(n \right)-4 a \! \left(n +1\right)-5 a \! \left(n +2\right)-2 a \! \left(n +3\right)+a \! \left(n +4\right)+a \! \left(n +5\right)+2 a \! \left(n +6\right), \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 128\)
\(\displaystyle a \! \left(7\right) = 303\)
\(\displaystyle a \! \left(8\right) = 703\)
\(\displaystyle a \! \left(9\right) = 1605\)
\(\displaystyle a \! \left(10\right) = 3610\)
\(\displaystyle a \! \left(n +7\right) = -2 a \! \left(n \right)-4 a \! \left(n +1\right)-5 a \! \left(n +2\right)-2 a \! \left(n +3\right)+a \! \left(n +4\right)+a \! \left(n +5\right)+2 a \! \left(n +6\right), \quad n \geq 11\)
Explicit Closed Form
\(\displaystyle -\frac{118538 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+4 Z^{6}+5 Z^{5}+2 Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +5}\right)}{19239}-\frac{597449 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+4 Z^{6}+5 Z^{5}+2 Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{38478}-\frac{897893 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+4 Z^{6}+5 Z^{5}+2 Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{3-n}\right)}{38478}-\frac{347669 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+4 Z^{6}+5 Z^{5}+2 Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{2-n}\right)}{19239}-\frac{79301 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+4 Z^{6}+5 Z^{5}+2 Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{12826}-\frac{9631 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+4 Z^{6}+5 Z^{5}+2 Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{38478}+\frac{119183 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}+4 Z^{6}+5 Z^{5}+2 Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{19239}-\left(\left\{\begin{array}{cc}-\frac{9}{4} & n =0 \\ -1 & n =1 \\ \frac{1}{2} & n =2 \\ 1 & n =3 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 60 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 60 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_{20}\! \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_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{22}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \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_{18}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \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_{16}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \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_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{42}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{43}\! \left(x \right)+F_{53}\! \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_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{48}\! \left(x \right)\\
\end{align*}\)