Av(1234, 1243, 1342, 1423, 2314, 2341)
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Generating Function
\(\displaystyle \frac{\left(-2 x^{2}+2 x -1\right) \sqrt{1-4 x}-2 x^{3}+2 x^{2}-2 x +1}{2 x \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 18, 54, 166, 527, 1724, 5781, 19770, 68677, 241616, 859003, 3081054, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x \right)^{2}+\left(2 x^{3}-2 x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{5}+2 x^{4}-6 x^{3}+7 x^{2}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{2 \left(9 n +10\right) a \! \left(1+n \right)}{n +5}-\frac{8 \left(2 n +5\right) a \! \left(n +2\right)}{n +5}+\frac{\left(26+7 n \right) a \! \left(n +3\right)}{n +5}+\frac{2}{n +5}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" 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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= y x\\ F_{28}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{31}\! \left(x , y\right) &= \frac{y F_{32}\! \left(x , y\right)-F_{32}\! \left(x , 1\right)}{-1+y}\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{28}\! \left(x , y\right) F_{8}\! \left(x \right)\\ \end{align*}\)