Av(1234, 1243, 1432, 2431)
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Generating Function
\(\displaystyle \frac{3 x^{7}-7 x^{6}+20 x^{5}-39 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 195, 571, 1622, 4505, 12310, 33247, 89049, 237090, 628528, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{4} F \! \left(x \right)-3 x^{7}+7 x^{6}-20 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 195\)
\(\displaystyle a \! \left(7\right) = 571\)
\(\displaystyle a \! \left(n +3\right) = -\frac{n^{3}}{6}+\frac{3 n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{10 n}{3}+4, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(66 \sqrt{5}-120\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-66 \sqrt{5}-120\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}-\frac{n^{3}}{6}+\frac{3 n^{2}}{2}\\-\frac{16 n}{3}-3 \,2^{n -1}+9 & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 181 rules.

Found on January 18, 2022.

Finding the specification took 12 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_{19}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{178}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{15}\! \left(x \right)+F_{175}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{40}\! \left(x \right) &= 0\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)+F_{62}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{60}\! \left(x \right) &= 0\\ F_{61}\! \left(x \right) &= 0\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \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_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{76}\! \left(x \right) &= 0\\ F_{77}\! \left(x \right) &= 0\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{82}\! \left(x \right)+F_{93}\! \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_{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_{33}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{70}\! \left(x \right)+F_{88}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{92}\! \left(x \right) &= 0\\ 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_{95}\! \left(x \right) &= F_{4}\! \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_{12}\! \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_{4}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{103}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{104}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{110}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{98}\! \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_{101}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{112}\! \left(x \right)+F_{116}\! \left(x \right)+F_{117}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{116}\! \left(x \right) &= 0\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{120}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{124}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{138}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{116}\! \left(x \right)+F_{117}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{139}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{151}\! \left(x \right)+F_{174}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{155}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{156}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{15}\! \left(x \right)+F_{157}\! \left(x \right)+F_{93}\! \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_{161}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{160}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{163}\! \left(x \right)+F_{70}\! \left(x \right)+F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{164}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{167}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{166}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{168}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{169}\! \left(x \right)+F_{173}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{170}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{172}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{166}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{168}\! \left(x \right)\\ F_{173}\! \left(x \right) &= 0\\ F_{174}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{177}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{179}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{180}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{2}\! \left(x \right)\\ \end{align*}\)