Av(13452, 13524, 13542, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 31452, 31524, 31542, 35124, 35142, 51234, 51243, 51324, 51342, 51423, 51432, 53124, 53142)
Generating Function
\(\displaystyle -\frac{6 x^{3}-4 x^{2}+4 x -1}{x^{4}-9 x^{3}+7 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 95, 359, 1340, 5018, 18846, 70805, 265925, 998586, 3749854, 14081688, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-9 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)+6 x^{3}-4 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 4 \right)} = - a{\left(n \right)} + 9 a{\left(n + 1 \right)} - 7 a{\left(n + 2 \right)} + 5 a{\left(n + 3 \right)}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 4 \right)} = - a{\left(n \right)} + 9 a{\left(n + 1 \right)} - 7 a{\left(n + 2 \right)} + 5 a{\left(n + 3 \right)}, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle -\frac{1306 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{61651}+\frac{16439 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{61651}-\frac{1317 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{61651}+\frac{1294 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{61651}\)
This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 43 rules.
Finding the specification took 65 seconds.
Copy 43 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_{16}\! \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_{25}\! \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_{16}\! \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_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{16}\! \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_{2}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{16}\! \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_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{16}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{16}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
\end{align*}\)