Av(12453, 12534, 12543, 13452, 13524, 13542, 14523, 14532, 21453, 21534, 21543, 23451, 23541, 24531, 31452, 31542, 32451, 32541, 41352, 42351)
Generating Function
\(\displaystyle \frac{6 x^{3}-6 x^{2}+5 x -1}{x^{6}-2 x^{5}-2 x^{4}+10 x^{3}-10 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 415, 1715, 7082, 29250, 120824, 499104, 2061705, 8516481, 35179810, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}-2 x^{5}-2 x^{4}+10 x^{3}-10 x^{2}+6 x -1\right) F \! \left(x \right)-6 x^{3}+6 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 6 \right)} = a{\left(n \right)} - 2 a{\left(n + 1 \right)} - 2 a{\left(n + 2 \right)} + 10 a{\left(n + 3 \right)} - 10 a{\left(n + 4 \right)} + 6 a{\left(n + 5 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 6 \right)} = a{\left(n \right)} - 2 a{\left(n + 1 \right)} - 2 a{\left(n + 2 \right)} + 10 a{\left(n + 3 \right)} - 10 a{\left(n + 4 \right)} + 6 a{\left(n + 5 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{1585 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4}}{182662}-\frac{1585 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +4}}{182662}-\frac{1585 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +4}}{182662}-\frac{1585 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +4}}{182662}-\frac{1585 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +4}}{182662}-\frac{1585 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +4}}{182662}+\frac{2209 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{91331}+\frac{2209 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{91331}+\frac{2209 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3}}{91331}+\frac{2209 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3}}{91331}+\frac{2209 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3}}{91331}+\frac{2209 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +3}}{91331}+\frac{23835 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{182662}+\frac{23835 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{182662}+\frac{23835 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{182662}+\frac{23835 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2}}{182662}+\frac{23835 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2}}{182662}+\frac{23835 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +2}}{182662}+\frac{2549 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{182662}+\frac{2549 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{182662}+\frac{2549 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{182662}+\frac{2549 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{182662}+\frac{2549 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1}}{182662}+\frac{2549 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +1}}{182662}+\frac{604 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n -1}}{91331}+\frac{604 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n -1}}{91331}+\frac{604 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n -1}}{91331}+\frac{604 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n -1}}{91331}+\frac{604 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n -1}}{91331}+\frac{604 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n -1}}{91331}+\frac{4098 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{91331}+\frac{4098 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{91331}+\frac{4098 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{91331}+\frac{4098 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{91331}+\frac{4098 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{91331}+\frac{4098 \mathit{RootOf} \left(Z^{6}-2 Z^{5}-2 Z^{4}+10 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n}}{91331}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 97 rules.
Finding the specification took 128 seconds.
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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_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{77}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{54}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{14}\! \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_{38}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{33}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{14}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{46}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{14}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{48}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{14}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{14}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{14}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{14}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{61}\! \left(x \right) &= 0\\
F_{62}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{54}\! \left(x \right)+F_{64}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{14}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{33}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{69}\! \left(x \right)+F_{7}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{14}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{14}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{41}\! \left(x \right)+F_{7}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{14}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{14}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{7}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{14}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{85}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{86}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{14}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{14}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{51}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{14}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{54}\! \left(x \right)+F_{7}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{14}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{11}\! \left(x \right)\\
\end{align*}\)