Av(1243, 1342, 4132, 4231)
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Generating Function
\(\displaystyle -\frac{13 x^{5}-35 x^{4}+42 x^{3}-26 x^{2}+8 x -1}{\left(x -1\right)^{3} \left(2 x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 199, 574, 1574, 4143, 10553, 26180, 63568, 151645, 356459, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{3} \left(2 x -1\right)^{3} F \! \left(x \right)+13 x^{5}-35 x^{4}+42 x^{3}-26 x^{2}+8 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-\frac{\left(-1+n \right) \left(n -4\right)}{2}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \frac{\left(n^{2}-n -8\right) 2^{n}}{8}+\frac{n^{2}}{2}+\frac{n}{2}+2\)
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This specification was found using the strategy pack "Point Placements" and has 73 rules.

Found on July 23, 2021.

Finding the specification took 5 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_{18}\! \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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \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_{13}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16} \left(x \right)^{2}\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{18}\! \left(x \right) &= x\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{18}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{28}\! \left(x \right) &= 0\\ F_{29}\! \left(x \right) &= F_{18}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{18}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{19}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{35}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{18}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{18}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{18}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{28}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{18}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{18}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{18}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{43}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{18}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{46}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{18}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{18}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{14}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{18}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{18}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{69}\! \left(x \right) &= 2 F_{28}\! \left(x \right)+F_{70}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{18}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{18}\! \left(x \right) F_{65}\! \left(x \right)\\ \end{align*}\)