Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 23514, 25134, 25143, 25314, 31245, 31254, 32145, 32154, 32514)
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Generating Function
\(\displaystyle -\frac{4 x^{4}+9 x^{3}-x^{2}-4 x +1}{\left(x -1\right) \left(4 x^{4}+9 x^{3}-2 x^{2}-4 x +1\right)}\)
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Counting Sequence
1, 1, 2, 6, 24, 95, 375, 1459, 5644, 21748, 83658, 321505, 1235037, 4743253, 18214918, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(4 x^{4}+9 x^{3}-2 x^{2}-4 x +1\right) F \! \left(x \right)+4 x^{4}+9 x^{3}-x^{2}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - 4 a{\left(n \right)} - 9 a{\left(n + 1 \right)} + 2 a{\left(n + 2 \right)} + 4 a{\left(n + 3 \right)} + 9, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{490361 \left(\underset{\alpha =\mathit{RootOf} \left(4 Z^{5}+5 Z^{4}-11 Z^{3}-2 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{514984}+\frac{3969745 \left(\underset{\alpha =\mathit{RootOf} \left(4 Z^{5}+5 Z^{4}-11 Z^{3}-2 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{2059936}-\frac{1056687 \left(\underset{\alpha =\mathit{RootOf} \left(4 Z^{5}+5 Z^{4}-11 Z^{3}-2 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{1029968}-\frac{275347 \left(\underset{\alpha =\mathit{RootOf} \left(4 Z^{5}+5 Z^{4}-11 Z^{3}-2 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{257492}+\frac{702389 \left(\underset{\alpha =\mathit{RootOf} \left(4 Z^{5}+5 Z^{4}-11 Z^{3}-2 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{2059936}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 131 rules.

Finding the specification took 131 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_{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_{38}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \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_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{24}\! \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_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= x^{2}\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \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_{22}\! \left(x \right)\\ F_{37}\! \left(x \right) &= 0\\ F_{38}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{18}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{15}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{15}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{15}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{15}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{65}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{15}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{15}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{15}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{15}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{15}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{15}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{15}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{18}\! \left(x \right)+F_{90}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{15}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{15}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{18}\! \left(x \right)+F_{73}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{15}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{100}\! \left(x \right) &= 0\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)+F_{106}\! \left(x \right)+F_{107}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{15}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{106}\! \left(x \right) &= 0\\ F_{107}\! \left(x \right) &= 0\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)+F_{18}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{15}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{114}\! \left(x \right) &= 0\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{117}\! \left(x \right)+F_{118}\! \left(x \right)+F_{18}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{15}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{117}\! \left(x \right) &= 0\\ F_{118}\! \left(x \right) &= 0\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{18}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{68}\! \left(x \right)\\ \end{align*}\)