Av(12345, 12354, 13245, 13254, 13425, 13452, 13524, 13542, 31245, 31254, 31425, 31452, 31524, 31542, 34125, 34152, 34512, 35124, 35142, 35412)
Generating Function
\(\displaystyle \frac{9 x^{5}-29 x^{4}+44 x^{3}-30 x^{2}+9 x -1}{\left(2 x -1\right) \left(3 x -1\right)^{2} \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 400, 1526, 5600, 19944, 69408, 237226, 799216, 2661308, 8776976, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(3 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)-9 x^{5}+29 x^{4}-44 x^{3}+30 x^{2}-9 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 3 \right)} = - 2 \left(n - 1\right) + 18 a{\left(n \right)} - 21 a{\left(n + 1 \right)} + 8 a{\left(n + 2 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 3 \right)} = - 2 \left(n - 1\right) + 18 a{\left(n \right)} - 21 a{\left(n + 1 \right)} + 8 a{\left(n + 2 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 2^{-1+n}+\frac{\left(n -3\right) 3^{n}}{6}+\frac{n}{2}+\frac{1}{2} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 89 rules.
Finding the specification took 97 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 89 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_{19}\! \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_{36}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{19}\! \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_{20}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \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_{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) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{19}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{19}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{19}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{19}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{37}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{19}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{19}\! \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_{1}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{19}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{37}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{19}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{19}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{19}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{19}\! \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_{16}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{19}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{19}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{71}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{19}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{19}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{71}\! \left(x \right)+F_{79}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{19}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{81}\! \left(x \right) &= 0\\
F_{82}\! \left(x \right) &= F_{19}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{53}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{71}\! \left(x \right)+F_{79}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= 0\\
\end{align*}\)