Av(1243, 1324, 1342, 1423, 2134)
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Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{2} \sqrt{-4 x +1}+5 x^{2}-4 x +1}{2 x^{2}}\)
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Counting Sequence
1, 1, 2, 6, 19, 62, 207, 704, 2431, 8502, 30056, 107236, 385662, 1396652, 5088865, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} F \left(x \right)^{2}+\left(-5 x^{2}+4 x -1\right) F \! \left(x \right)+x^{3}+2 x^{2}-3 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(2+n \right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{4+n}+\frac{\left(11+5 n \right) a \! \left(n +1\right)}{4+n}, \quad n \geq 3\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 17 rules.

Found on July 23, 2021.

Finding the specification took 1 seconds.

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