Av(12354, 12453, 13254, 13452, 14253, 14352, 21354, 21453, 23154, 23451, 24153, 24351, 31254, 31452, 32154, 32451, 34152, 34251, 41253, 41352, 42153, 42351, 43152, 43251, 51243, 51342, 52143, 52341, 53142, 53241)
Generating Function
\(\displaystyle \frac{6 x^{4}+3 x^{3}+5 x^{2}-5 x +1}{\left(3 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 90, 324, 1134, 3888, 13122, 43740, 144342, 472392, 1535274, 4960116, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(3 x -1\right)^{2} F \! \left(x \right)+6 x^{4}+3 x^{3}+5 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 2 \right)} = - 9 a{\left(n \right)} + 6 a{\left(n + 1 \right)}, \quad n \geq 5\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 2 \right)} = - 9 a{\left(n \right)} + 6 a{\left(n + 1 \right)}, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{2 \,3^{n} n}{27} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 81 rules.
Finding the specification took 68 seconds.
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\(\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_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= 0\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{23}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{14}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{14}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{14}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= x^{2}\\
F_{44}\! \left(x \right) &= x^{2}\\
F_{45}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{39}\! \left(x \right)+F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{14}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{14}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{14}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{49}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{14}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{14}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{14}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{14}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{14}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{44}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{14}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{47}\! \left(x \right)+F_{71}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{14}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{77}\! \left(x \right)\\
\end{align*}\)