Av(1234, 1243, 1342, 2134, 2314, 3412)
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Generating Function
\(\displaystyle -\frac{4 x^{5}-11 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}\)
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Counting Sequence
1, 1, 2, 6, 18, 48, 115, 255, 537, 1095, 2192, 4348, 8596, 16994, 33649, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+4 x^{5}-11 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 48\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{3}-9 n^{2}+2 n -24\right)}{24}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle -1+2^{n +1}-\frac{n^{2}}{24}-\frac{7 n}{4}-\frac{n^{3}}{4}+\frac{n^{4}}{24}\)
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This specification was found using the strategy pack "Point Placements" and has 30 rules.

Found on July 23, 2021.

Finding the specification took 7 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_{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_{12}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9} \left(x \right)^{3}\\ 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_{14}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\ \end{align*}\)