Av(1324, 2413, 4231)
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Generating Function
\(\displaystyle -\frac{x^{6}-4 x^{5}+23 x^{4}-36 x^{3}+25 x^{2}-8 x +1}{\left(x -1\right) \left(2 x -1\right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 21, 73, 242, 762, 2290, 6610, 18434, 49922, 131842, 340738, 864258, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x -1\right)^{4} F \! \left(x \right)+x^{6}-4 x^{5}+23 x^{4}-36 x^{3}+25 x^{2}-8 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 242\)
\(\displaystyle a \! \left(n +4\right) = -16 a \! \left(n \right)+32 a \! \left(n +1\right)-24 a \! \left(n +2\right)+8 a \! \left(n +3\right)+2, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2+\frac{\left(5 n^{3}-27 n^{2}+136 n -204\right) 2^{n}}{192} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point And Row Placements" and has 67 rules.

Found on July 23, 2021.

Finding the specification took 6 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_{13}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{27}\! \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_{14}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{23}\! \left(x \right) &= 0\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{13}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{13}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{35}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{13}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{13}\! \left(x \right) F_{38}\! \left(x \right) F_{44}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{13}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{13}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{13}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{51}\! \left(x \right)+F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{13}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{13}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{13}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{13}\! \left(x \right) F_{44}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= 2 F_{66}\! \left(x \right)+F_{1}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{21} \left(x \right)^{2} F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{13}\! \left(x \right) F_{62}\! \left(x \right)\\ \end{align*}\)