Av(1243, 1324, 1342)
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Generating Function
\(\displaystyle \frac{-5 \sqrt{-4 x +1}\, x -2 x^{2}+\sqrt{-4 x +1}+13 x -3}{2 x^{2}+8 x -2}\)
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Counting Sequence
1, 1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+4 x -1\right) F \left(x \right)^{2}+\left(2 x^{2}-13 x +3\right) F \! \left(x \right)+x^{2}+8 x -2 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +4\right) = \frac{10 \left(1+2 n \right) a \! \left(n \right)}{n +4}+\frac{\left(20+71 n \right) a \! \left(n +1\right)}{n +4}-\frac{\left(86+55 n \right) a \! \left(n +2\right)}{n +4}+\frac{\left(36+13 n \right) a \! \left(n +3\right)}{n +4}, \quad n \geq 5\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 23 rules.

Found on July 23, 2021.

Finding the specification took 11 seconds.

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