Av(1234, 1243, 1342, 2314)
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Generating Function
\(\displaystyle -\frac{\left(x +\left(x -1\right) \sqrt{-4 x +1}\right) \left(x -1\right)}{4 x^{3}-8 x^{2}+6 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 68, 234, 816, 2882, 10292, 37098, 134776, 492928, 1813280, 6704036, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{3}-8 x^{2}+6 x -1\right) F \left(x \right)^{2}+2 x \left(x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +5\right) = \frac{8 \left(1+2 n \right) a \! \left(n \right)}{n +5}-\frac{4 \left(20+13 n \right) a \! \left(n +1\right)}{n +5}+\frac{4 \left(40+17 n \right) a \! \left(n +2\right)}{n +5}-\frac{6 \left(23+7 n \right) a \! \left(n +3\right)}{n +5}+\frac{\left(46+11 n \right) a \! \left(n +4\right)}{n +5}, \quad n \geq 5\)
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This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 21 rules.

Found on July 23, 2021.

Finding the specification took 4 seconds.

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