Av(12354, 12453, 12543, 13254, 13452, 13542, 14253, 14352, 14532, 15243, 15342, 15432, 21354, 21453, 21543, 23154, 23451, 23541, 24153, 24351, 24531, 25143, 25341, 25431, 31254, 31452, 31542, 32154, 32451, 32541, 34152, 34251, 35142, 35241, 41253, 41352, 42153, 42351, 43152, 43251)
Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(2 x^{5}-3 x^{4}+2 x^{3}-2 x^{2}+3 x -1\right)}{3 x^{4}-8 x^{3}+8 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 80, 252, 794, 2522, 8034, 25590, 81472, 259346, 825572, 2628098, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(-3 x^{4}+8 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(2 x^{5}-3 x^{4}+2 x^{3}-2 x^{2}+3 x -1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 80\)
\(\displaystyle a(6) = 252\)
\(\displaystyle a{\left(n + 4 \right)} = - 3 a{\left(n \right)} + 8 a{\left(n + 1 \right)} - 8 a{\left(n + 2 \right)} + 5 a{\left(n + 3 \right)}, \quad n \geq 7\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 80\)
\(\displaystyle a(6) = 252\)
\(\displaystyle a{\left(n + 4 \right)} = - 3 a{\left(n \right)} + 8 a{\left(n + 1 \right)} - 8 a{\left(n + 2 \right)} + 5 a{\left(n + 3 \right)}, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{2350 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-8 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{2-n}\right)}{6729}+\frac{844 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-8 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{6729}+\frac{2030 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-8 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{6729}-\frac{142 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-8 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{6729}+\frac{\left(\left\{\begin{array}{cc}\frac{5}{9} & n =0 \\ \frac{1}{3} & n =1 \\ 2 & n =2 \\ 0 & \text{otherwise} \end{array}\right.\right)}{3}\)
This specification was found using the strategy pack "Point Placements" and has 96 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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{16}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{29}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{16}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{16}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{57}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{16}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{62}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{16}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{62}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{16}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{70}\! \left(x \right)+F_{78}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{16}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{16}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{16}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{16}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{70}\! \left(x \right)+F_{75}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{16}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{16}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{16}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{70}\! \left(x \right)+F_{78}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{16}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{16}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{89}\! \left(x \right)\\
\end{align*}\)