Av(1243, 2314, 2341, 4231)
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Generating Function
\(\displaystyle \frac{3 x^{7}-16 x^{6}+49 x^{5}-77 x^{4}+68 x^{3}-34 x^{2}+9 x -1}{\left(2 x -1\right)^{3} \left(-1+x \right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 192, 543, 1462, 3782, 9472, 23109, 55190, 129524, 299616, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{3} \left(-1+x \right)^{4} F \! \left(x \right)-3 x^{7}+16 x^{6}-49 x^{5}+77 x^{4}-68 x^{3}+34 x^{2}-9 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 192\)
\(\displaystyle a \! \left(7\right) = 543\)
\(\displaystyle a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-\frac{\left(n -3\right) \left(n -2\right) \left(n -1\right)}{6}, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(3 n^{2}+39 n -258\right) 2^{n}}{48}+\frac{n^{3}}{6}+\frac{n^{2}}{2}+\frac{10 n}{3}+6 & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 158 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \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_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ 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_{12}\! \left(x \right) F_{16}\! \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_{12}\! \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_{12}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{25}\! \left(x \right) &= 0\\ F_{26}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{25}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{12}\! \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_{36}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{60}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{68}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{12}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{76}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{12}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{12}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{82}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{83}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{12}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{12}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{12}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{12}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{99}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{100}\! \left(x \right)+F_{114}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= 3 F_{25}\! \left(x \right)+F_{108}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{117}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{12}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{12}\! \left(x \right) F_{120}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{127}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{12}\! \left(x \right) F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{12}\! \left(x \right) F_{121}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{148}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{134}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{12}\! \left(x \right) F_{131}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{12}\! \left(x \right) F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{12}\! \left(x \right) F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{141}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{12}\! \left(x \right) F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{143}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{153}\! \left(x \right)+F_{157}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{12}\! \left(x \right) F_{150}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{12}\! \left(x \right) F_{154}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{156}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{122}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{148}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{12}\! \left(x \right) F_{128}\! \left(x \right)\\ \end{align*}\)