Av(12453, 12543, 13452, 13542, 14352, 14532, 15342, 15432, 21453, 21543, 23451, 23541, 24351, 24531, 25341, 25431, 31452, 31542, 32451, 32541)
Generating Function
\(\displaystyle -\frac{2 x^{3}+4 x^{2}-5 x +1}{4 x^{4}-8 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 416, 1720, 7088, 29168, 119968, 493344, 2028672, 8341952, 34302208, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{4}-8 x^{2}+6 x -1\right) F \! \left(x \right)+2 x^{3}+4 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n \right)} = 2 a{\left(n + 2 \right)} - \frac{3 a{\left(n + 3 \right)}}{2} + \frac{a{\left(n + 4 \right)}}{4}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n \right)} = 2 a{\left(n + 2 \right)} - \frac{3 a{\left(n + 3 \right)}}{2} + \frac{a{\left(n + 4 \right)}}{4}, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle -\frac{39 \left(\underset{\alpha =\mathit{RootOf} \left(4 Z^{4}-8 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{43}-\frac{24 \left(\underset{\alpha =\mathit{RootOf} \left(4 Z^{4}-8 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{43}+\frac{143 \left(\underset{\alpha =\mathit{RootOf} \left(4 Z^{4}-8 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{86}-\frac{29 \left(\underset{\alpha =\mathit{RootOf} \left(4 Z^{4}-8 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{86}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 42 rules.
Finding the specification took 65 seconds.
Copy 42 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_{15}\! \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_{38}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{15}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{15}\! \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_{15}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{25}\! \left(x \right) &= 0\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{37}\! \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_{7}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 0\\
F_{37}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{15}\! \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_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
\end{align*}\)