Av(12345, 12354, 12435, 12453, 12534, 12543, 13425, 13452, 13524, 13542, 14523, 14532, 21345, 21354, 21435, 21453, 21534, 21543, 23415, 23514, 24513, 31245, 31254, 31425, 31524, 32415, 32514, 41235, 41325, 42315)
Generating Function
\(\displaystyle \frac{4 x^{3}-5 x^{2}+4 x -1}{2 x^{4}+8 x^{3}-8 x^{2}+5 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 90, 310, 1034, 3458, 11678, 39618, 134398, 455386, 1542046, 5221562, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}+8 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)-4 x^{3}+5 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 4 \right)} = 2 a{\left(n \right)} + 8 a{\left(n + 1 \right)} - 8 a{\left(n + 2 \right)} + 5 a{\left(n + 3 \right)}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 4 \right)} = 2 a{\left(n \right)} + 8 a{\left(n + 1 \right)} - 8 a{\left(n + 2 \right)} + 5 a{\left(n + 3 \right)}, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \frac{460 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}+8 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{19947}+\frac{1151 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}+8 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{19947}+\frac{3244 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}+8 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{19947}+\frac{107 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}+8 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{19947}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 143 rules.
Finding the specification took 165 seconds.
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Copy 143 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_{36}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{17}\! \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_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 0\\
F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \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_{35}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{24}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{16}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{41}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{16}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{16}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{16}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{16}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{16}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{51}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{34}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{16}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{16}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{24}\! \left(x \right)+F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{16}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{78}\! \left(x \right)+F_{84}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{16}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{88}\! \left(x \right)+F_{89}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{16}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{16}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{92}\! \left(x \right) &= 0\\
F_{93}\! \left(x \right) &= F_{16}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{16}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{99}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{100}\! \left(x \right)+F_{101}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{16}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{104}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{100}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{103}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{24}\! \left(x \right)+F_{88}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{114}\! \left(x \right)+F_{116}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{116}\! \left(x \right) &= 0\\
F_{117}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{24}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{123}\! \left(x \right)+F_{24}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{24}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{24}\! \left(x \right)+F_{84}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{24}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{141}\! \left(x \right)+F_{24}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{40}\! \left(x \right)\\
\end{align*}\)