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, 23415, 23514, 24315, 24513, 25314, 25413)
Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(2 x -1\right) \left(x^{2}+2 x -1\right)}{6 x^{6}-6 x^{5}+6 x^{4}+6 x^{3}-11 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 90, 324, 1128, 3864, 13152, 44724, 152220, 518628, 1768188, 6030156, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(6 x^{6}-6 x^{5}+6 x^{4}+6 x^{3}-11 x^{2}+6 x -1\right) F \! \left(x \right)-\left(x -1\right) \left(2 x -1\right) \left(x^{2}+2 x -1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a{\left(n + 6 \right)} = 6 a{\left(n \right)} - 6 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)} + 6 a{\left(n + 3 \right)} - 11 a{\left(n + 4 \right)} + 6 a{\left(n + 5 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a{\left(n + 6 \right)} = 6 a{\left(n \right)} - 6 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)} + 6 a{\left(n + 3 \right)} - 11 a{\left(n + 4 \right)} + 6 a{\left(n + 5 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{199896 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4}}{854053}-\frac{199896 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +4}}{854053}-\frac{199896 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +4}}{854053}-\frac{199896 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +4}}{854053}-\frac{199896 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +4}}{854053}-\frac{199896 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +4}}{854053}+\frac{6066 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3}}{854053}+\frac{6066 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3}}{854053}+\frac{6066 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3}}{854053}+\frac{6066 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +3}}{854053}+\frac{6066 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{854053}+\frac{6066 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{854053}-\frac{93679 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{854053}-\frac{93679 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{854053}-\frac{93679 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{854053}-\frac{93679 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2}}{854053}-\frac{93679 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2}}{854053}-\frac{93679 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +2}}{854053}-\frac{269122 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{854053}-\frac{269122 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{854053}-\frac{269122 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{854053}-\frac{269122 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{854053}-\frac{269122 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1}}{854053}-\frac{269122 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +1}}{854053}+\frac{5350 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n -1}}{854053}+\frac{5350 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n -1}}{854053}+\frac{5350 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n -1}}{854053}+\frac{5350 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n -1}}{854053}+\frac{5350 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n -1}}{854053}+\frac{5350 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n -1}}{854053}+\frac{249025 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{854053}+\frac{249025 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{854053}+\frac{249025 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{854053}+\frac{249025 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{854053}+\frac{249025 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{854053}+\frac{249025 \mathit{RootOf} \left(6 Z^{6}-6 Z^{5}+6 Z^{4}+6 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n}}{854053}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 66 rules.
Finding the specification took 86 seconds.
Copy 66 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_{18}\! \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_{32}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \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_{11}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 0\\
F_{25}\! \left(x \right) &= F_{18}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{33}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \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_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{18}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{18}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{18}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{18}\! \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_{42}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{52}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{18}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{18}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{18}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{18}\! \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_{11}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{51}\! \left(x \right)\\
\end{align*}\)