Av(1243, 1324, 1342, 1423, 3412)
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Generating Function
\(\displaystyle -\frac{4 x^{6}-19 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 19, 56, 153, 396, 992, 2445, 5997, 14743, 36471, 90939, 228610, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{6}-19 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 153\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{\left(n +1\right) \left(n^{2}+2 n -6\right)}{6}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{\left(-3 \sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(3 \sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}-\frac{n^{3}}{6}-\frac{n^{2}}{2}-\frac{4 n}{3}+2^{n +1}-2\)
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This specification was found using the strategy pack "Point Placements" and has 34 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_{13}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{15}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \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_{9}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{25}\! \left(x \right) &= 0\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{13}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{13}\! \left(x \right) F_{29}\! \left(x \right)\\ \end{align*}\)