Av(1234, 1243, 1342, 2341)
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Generating Function
\(\displaystyle \frac{4 x -1-x \sqrt{-4 x +1}}{4 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x -1\right) F \left(x \right)^{2}+\left(-8 x +2\right) F \! \left(x \right)+x^{2}+4 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(n +1\right) = \frac{2 \left(-1+2 n \right) a \! \left(n \right)}{n}, \quad n \geq 2\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 16 rules.

Found on January 20, 2022.

Finding the specification took 1 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , 1, y\right)\\ F_{9}\! \left(x , y , z\right) &= F_{10}\! \left(x , y z , z\right)\\ F_{10}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y , z\right)+F_{13}\! \left(x , y , z\right)\\ F_{11}\! \left(x , y , z\right) &= F_{10}\! \left(x , y , z\right) F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= y x\\ F_{13}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{14}\! \left(x , y , z\right)\\ F_{14}\! \left(x , y , z\right) &= \frac{-z F_{9}\! \left(x , 1, z\right)+y F_{9}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\ F_{15}\! \left(x \right) &= x\\ \end{align*}\)

This specification was found using the strategy pack "Row Placements Tracked Fusion Expand Verified" and has 23 rules.

Found on February 08, 2022.

Finding the specification took 3 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{15}\! \left(x , y\right)+F_{15}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= y x\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{21}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{22}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ \end{align*}\)