Av(13452, 13524, 13542, 14352, 14532, 15324, 15342, 15432, 31452, 31524, 31542, 35124, 35142, 51324, 51342, 51432, 53124, 53142)
Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(6 x^{3}-10 x^{2}+6 x -1\right)}{14 x^{4}-28 x^{3}+22 x^{2}-8 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 102, 428, 1768, 7248, 29644, 121208, 495688, 2027488, 8293576, 33926224, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(14 x^{4}-28 x^{3}+22 x^{2}-8 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(6 x^{3}-10 x^{2}+6 x -1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - 14 a{\left(n \right)} + 28 a{\left(n + 1 \right)} - 22 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)}, \quad n \geq 5\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - 14 a{\left(n \right)} + 28 a{\left(n + 1 \right)} - 22 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)}, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\sqrt{2}+4\right) \left(\frac{1}{2}-\frac{i \sqrt{7+14 \sqrt{2}}}{14}\right)^{-n}}{28}+\\\frac{\left(\sqrt{2}+4\right) \left(\frac{1}{2}+\frac{i \sqrt{7+14 \sqrt{2}}}{14}\right)^{-n}}{28}-\\\frac{\left(\left(\frac{1}{2}-\frac{\sqrt{-7+14 \sqrt{2}}}{14}\right)^{-n}+\left(\frac{1}{2}+\frac{\sqrt{-7+14 \sqrt{2}}}{14}\right)^{-n}\right) \left(\sqrt{2}-4\right)}{28} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 39 rules.
Finding the specification took 78 seconds.
Copy 39 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_{12}\! \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_{13}\! \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_{12}\! \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_{6}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\
\end{align*}\)