Av(1234, 1243, 1342, 2314, 2341)
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Generating Function
\(\displaystyle \frac{\left(2-3 x \right) \sqrt{-4 x +1}-2 x^{2}+5 x -2}{2 \left(x -1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 19, 61, 199, 661, 2234, 7668, 26674, 93858, 333524, 1195288, 4315468, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{2} F \left(x \right)^{2}+\left(x -1\right) \left(x -2\right) \left(2 x -1\right) F \! \left(x \right)+x^{3}+4 x^{2}-6 x +2 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = \frac{3 \left(1+2 n \right) a \! \left(n \right)}{n +4}-\frac{\left(32+23 n \right) a \! \left(1+n \right)}{2 \left(n +4\right)}+\frac{\left(34+13 n \right) a \! \left(n +2\right)}{2 n +8}, \quad n \geq 4\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 31 rules.

Found on July 23, 2021.

Finding the specification took 8 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_{25}\! \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_{22}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= \frac{y F_{18}\! \left(x , y\right)-F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{11}\! \left(x \right) F_{27}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\ F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{11}\! \left(x \right) F_{28}\! \left(x \right)\\ \end{align*}\)