Av(1234, 1342, 2341)
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Generating Function
\(\displaystyle \frac{-9 \left(x -\frac{1}{3}\right)^{2} \sqrt{-4 x +1}-4 x^{3}+17 x^{2}-8 x +1}{8 x^{3}-2 x^{2}}\)
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Counting Sequence
1, 1, 2, 6, 21, 78, 297, 1144, 4433, 17238, 67184, 262276, 1025202, 4011660, 15712335, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(4 x -1\right) F \left(x \right)^{2}+\left(4 x -1\right) \left(x^{2}-4 x +1\right) F \! \left(x \right)+x^{3}+12 x^{2}-7 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(2+n \right) = -\frac{6 \left(1+2 n \right) a \! \left(n \right)}{4+n}+\frac{\left(17+7 n \right) a \! \left(n +1\right)}{4+n}, \quad n \geq 3\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 26 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_{3}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{23}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\ 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_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\ F_{19}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{21}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= y x\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{24}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ \end{align*}\)