Av(12354, 12453, 12543, 13452, 13542, 14532, 21354, 21453, 21543, 23451, 23541, 24531, 31254, 31452, 31542, 32451, 32541, 41253, 41352, 42351)
Generating Function
\(\displaystyle \frac{7 x^{5}-21 x^{4}+31 x^{3}-23 x^{2}+8 x -1}{\left(x -1\right) \left(13 x^{4}-27 x^{3}+22 x^{2}-8 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 407, 1625, 6433, 25402, 100273, 395905, 1563458, 6174898, 24388993, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(13 x^{4}-27 x^{3}+22 x^{2}-8 x +1\right) F \! \left(x \right)-7 x^{5}+21 x^{4}-31 x^{3}+23 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 4 \right)} = - 13 a{\left(n \right)} + 27 a{\left(n + 1 \right)} - 22 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)} - 1, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 4 \right)} = - 13 a{\left(n \right)} + 27 a{\left(n + 1 \right)} - 22 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)} - 1, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{5512 \left(\underset{\alpha =\mathit{RootOf} \left(13 Z^{5}-40 Z^{4}+49 Z^{3}-30 Z^{2}+9 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{331}+\frac{11682 \left(\underset{\alpha =\mathit{RootOf} \left(13 Z^{5}-40 Z^{4}+49 Z^{3}-30 Z^{2}+9 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{331}-\frac{9999 \left(\underset{\alpha =\mathit{RootOf} \left(13 Z^{5}-40 Z^{4}+49 Z^{3}-30 Z^{2}+9 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{331}+\frac{4037 \left(\underset{\alpha =\mathit{RootOf} \left(13 Z^{5}-40 Z^{4}+49 Z^{3}-30 Z^{2}+9 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{331}-\frac{539 \left(\underset{\alpha =\mathit{RootOf} \left(13 Z^{5}-40 Z^{4}+49 Z^{3}-30 Z^{2}+9 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{331}+\left(\left\{\begin{array}{cc}\frac{7}{13} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 111 rules.
Finding the specification took 123 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_{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_{7}\! \left(x \right)+F_{8}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \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_{68}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{14}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{14}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{58}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{14}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{14}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{14}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{14}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{52}\! \left(x \right)+F_{56}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{14}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{14}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{14}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{51}\! \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_{64}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{62}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{14}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{58}\! \left(x \right)+F_{66}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{14}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{7}\! \left(x \right)+F_{70}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{14}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{14}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{43}\! \left(x \right)+F_{7}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{14}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{7}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{14}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{7}\! \left(x \right)+F_{84}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{14}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{14}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{7}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{14}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{43}\! \left(x \right)+F_{7}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{14}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{14}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{7}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{109}\! \left(x \right)+F_{58}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{88}\! \left(x \right)\\
\end{align*}\)