Av(1243, 1342, 1432, 2413, 2431, 3142, 4132)
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Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{3} \sqrt{-4 x +1}-2 x^{4}+x^{3}-3 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 17, 48, 142, 444, 1451, 4890, 16832, 58831, 208067, 742966, 2674518, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x \right)^{2}+\left(2 x^{4}-x^{3}+3 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x^{3}+2 x^{2}-3 x +1\right) \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 17\)
\(\displaystyle a \! \left(5\right) = 48\)
\(\displaystyle a \! \left(n +1\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{\left(n -1\right) \left(3 n^{2}-8 n -4\right)}{2 \left(n +2\right)}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements" and has 19 rules.

Found on July 23, 2021.

Finding the specification took 1 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9} \left(x \right)^{3}\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ \end{align*}\)