Av(1243, 1342, 1423, 2134, 2314, 2341, 3124)
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Generating Function
\(\displaystyle \frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{4}+x^{3}-x +1\right)}{2 x \left(x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 17, 48, 145, 461, 1518, 5124, 17619, 61468, 216986, 773532, 2780654, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{2} F \left(x \right)^{2}+\left(x -1\right) \left(x^{4}+x^{3}-x +1\right) F \! \left(x \right)+\left(x^{4}+x^{3}-x +1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 17\)
\(\displaystyle a \! \left(5\right) = 48\)
\(\displaystyle a \! \left(6\right) = 145\)
\(\displaystyle a \! \left(7\right) = 461\)
\(\displaystyle a \! \left(8\right) = 1518\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(2 n -5\right) a \! \left(n \right)}{7+n}+\frac{\left(n -9\right) a \! \left(1+n \right)}{7+n}+\frac{4 n a \! \left(n +2\right)}{7+n}+\frac{\left(3 n +13\right) a \! \left(n +3\right)}{7+n}-\frac{\left(41+9 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(3 n +17\right) a \! \left(n +5\right)}{7+n}, \quad n \geq 9\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 36 rules.

Found on January 20, 2022.

Finding the specification took 6 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_{3}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \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_{1}\! \left(x \right)+F_{20}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= \frac{y F_{11}\! \left(x , y\right)-F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{19}\! \left(x , y\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) &= F_{16}\! \left(x , y\right)\\ 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_{14}\! \left(x , y\right)-F_{14}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x \right) &= x\\ F_{19}\! \left(x , y\right) &= y x\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{18}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{18}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{18}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\ \end{align*}\)