Av(1243, 1324, 1342, 2134, 3124)
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Generating Function
\(\displaystyle \frac{-\left(x^{2}-x +1\right)^{2} \sqrt{1-4 x}+3 x^{4}-4 x^{3}+7 x^{2}-4 x +1}{2 \left(x^{3}+x +1\right) x^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 61, 201, 678, 2328, 8107, 28564, 101642, 364755, 1318581, 4797204, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+x +1\right) x^{2} F \left(x \right)^{2}+\left(-3 x^{4}+4 x^{3}-7 x^{2}+4 x -1\right) F \! \left(x \right)+x^{4}-2 x^{3}+4 x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(n +6\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{3 \left(11+3 n \right) a \! \left(2+n \right)}{8+n}-\frac{\left(16+5 n \right) a \! \left(n +1\right)}{8+n}-\frac{2 \left(11+n \right) a \! \left(n +3\right)}{8+n}+\frac{15 a \! \left(n +4\right)}{8+n}+\frac{\left(23+4 n \right) a \! \left(n +5\right)}{8+n}, \quad n \geq 6\)

This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 24 rules.

Found on July 23, 2021.

Finding the specification took 2 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= \frac{y F_{11}\! \left(x , y\right)-F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{20}\! \left(x , y\right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{22}\! \left(x , y\right) &= \frac{y F_{15}\! \left(x , y\right)-F_{15}\! \left(x , 1\right)}{-1+y}\\ F_{23}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\ \end{align*}\)