Av(12453, 12543, 13452, 13542, 21453, 21543, 23451, 23541, 31452, 31542, 32451, 32541)
Generating Function
\(\displaystyle \frac{2 x^{2}-5 x +1}{6 x^{2}-6 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 108, 504, 2376, 11232, 53136, 251424, 1189728, 5629824, 26640576, 126064512, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(6 x^{2}-6 x +1\right) F \! \left(x \right)-2 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a{\left(n + 2 \right)} = - 6 a{\left(n \right)} + 6 a{\left(n + 1 \right)}, \quad n \geq 3\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a{\left(n + 2 \right)} = - 6 a{\left(n \right)} + 6 a{\left(n + 1 \right)}, \quad n \geq 3\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(2-\sqrt{3}\right) \left(\frac{1}{2}-\frac{\sqrt{3}}{6}\right)^{-n}}{6}+\frac{\left(\frac{1}{2}+\frac{\sqrt{3}}{6}\right)^{-n} \left(2+\sqrt{3}\right)}{6} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 44 rules.
Finding the specification took 104 seconds.
Copy 44 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_{40}\! \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_{31}\! \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_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{15}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{15}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)+F_{38}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{15}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \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_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{6}\! \left(x \right)\\
\end{align*}\)