Av(1234, 1243, 1324, 1342, 2314, 2341, 3124, 4123)
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Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{2} \sqrt{-4 x +1}+2 x^{4}+x^{2}-2 x +1}{2 x \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 16, 45, 136, 434, 1436, 4869, 16804, 58795, 208022, 742911, 2674452, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x \right)^{2}-\left(2 x^{4}+x^{2}-2 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{7}+x^{5}-x^{4}-3 x^{3}+6 x^{2}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 16\)
\(\displaystyle a \! \left(5\right) = 45\)
\(\displaystyle a \! \left(1+n \right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{3 n^{2}-7 n -2}{n +2}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 38 rules.

Found on July 23, 2021.

Finding the specification took 11 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_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{14}\! \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_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{11}\! \left(x \right)\\ F_{10}\! \left(x , y\right) &= \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}\\ F_{11}\! \left(x \right) &= x\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x , 1\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{20}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x , y\right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{30}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{30}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= \frac{y F_{23}\! \left(x , y\right)-F_{23}\! \left(x , 1\right)}{-1+y}\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= \frac{y F_{18}\! \left(x , y\right)-F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{37}\! \left(x \right) &= F_{27} \left(x \right)^{2} F_{11}\! \left(x \right)\\ \end{align*}\)