Av(1234, 1342, 2341, 2413, 4132)
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Generating Function
\(\displaystyle -\frac{4 x^{7}-4 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 19, 53, 132, 308, 693, 1527, 3322, 7166, 15363, 32777, 69648, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+4 x^{7}-4 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 132\)
\(\displaystyle a \! \left(7\right) = 308\)
\(\displaystyle a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{17 n}{2}+22, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(n +3\right) 2^{n}}{4}+\frac{n^{2}}{2}-\frac{13 n}{2}+9 & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 169 rules.

Found on January 18, 2022.

Finding the specification took 3 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{38}\! \left(x \right) &= 0\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{38}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{38}\! \left(x \right)+F_{50}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{38}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{70}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{137}\! \left(x \right)+F_{38}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{100}\! \left(x \right) &= 2 F_{38}\! \left(x \right)+F_{101}\! \left(x \right)+F_{105}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{104}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{100}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= 3 F_{38}\! \left(x \right)+F_{116}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{122}\! \left(x \right) &= 0\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{125}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= 2 F_{38}\! \left(x \right)+F_{131}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{130}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{146}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{151}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{147}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{156}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{143}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{152}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{160}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{165}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{163}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{164}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{161}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{168}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{165}\! \left(x \right)\\ \end{align*}\)