Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 23451, 23541, 24135, 24153, 24351, 24531, 25134, 25143, 25341, 25431, 31245, 31254, 31452, 31542, 32145, 32154, 32451, 32541, 34152, 34251, 35142, 35241, 41253, 41352, 42153, 42351, 43152, 43251)
Generating Function
\(\displaystyle \frac{x^{8}+6 x^{7}+11 x^{6}+11 x^{5}+11 x^{4}-3 x^{3}-x^{2}-x +1}{\left(x^{4}+3 x^{3}+x^{2}+x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 60, 150, 399, 992, 2412, 5880, 14135, 33612, 79534, 187068, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}+3 x^{3}+x^{2}+x -1\right)^{2} F \! \left(x \right)-x^{8}-6 x^{7}-11 x^{6}-11 x^{5}-11 x^{4}+3 x^{3}+x^{2}+x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 60\)
\(\displaystyle a(6) = 150\)
\(\displaystyle a(7) = 399\)
\(\displaystyle a(8) = 992\)
\(\displaystyle a{\left(n + 8 \right)} = - a{\left(n \right)} - 6 a{\left(n + 1 \right)} - 11 a{\left(n + 2 \right)} - 8 a{\left(n + 3 \right)} - 5 a{\left(n + 4 \right)} + 4 a{\left(n + 5 \right)} + a{\left(n + 6 \right)} + 2 a{\left(n + 7 \right)}, \quad n \geq 9\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 60\)
\(\displaystyle a(6) = 150\)
\(\displaystyle a(7) = 399\)
\(\displaystyle a(8) = 992\)
\(\displaystyle a{\left(n + 8 \right)} = - a{\left(n \right)} - 6 a{\left(n + 1 \right)} - 11 a{\left(n + 2 \right)} - 8 a{\left(n + 3 \right)} - 5 a{\left(n + 4 \right)} + 4 a{\left(n + 5 \right)} + a{\left(n + 6 \right)} + 2 a{\left(n + 7 \right)}, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \frac{\left(209 n +209\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+3 Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -2}\right)}{2243}+\frac{\left(-181 n -209\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+3 Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{2243}+\frac{\left(38 n -627\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+3 Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{2243}-\frac{209 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+3 Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{2243}+\frac{\left(-83 n -209\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+3 Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{2243}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 81 rules.
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\(\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_{16}\! \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_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \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_{4}\! \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_{73}\! \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_{38}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \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_{33}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{29}\! \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) &= 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{50}\! \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_{32}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{54}\! \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_{45}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{60}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{78}\! \left(x \right) &= 0\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{67}\! \left(x \right)\\
\end{align*}\)