Av(1342, 1423, 2314, 3124)
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Generating Function
\(\displaystyle -\frac{4 \left(x -\frac{1}{2}\right) \left(x -1\right)^{2} \left(\left(x -\frac{1}{2}\right) \left(x -1\right)^{2} \sqrt{1-4 x}-2 x^{3}+\frac{5 x^{2}}{2}-2 x +\frac{1}{2}\right)}{8 x^{7}-34 x^{6}+72 x^{5}-80 x^{4}+50 x^{3}-16 x^{2}+2 x}\)
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Counting Sequence
1, 1, 2, 6, 20, 66, 215, 702, 2319, 7772, 26415, 90926, 316573, 1113531, 3952955, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(4 x^{6}-17 x^{5}+36 x^{4}-40 x^{3}+25 x^{2}-8 x +1\right) F \left(x \right)^{2}-\left(2 x -1\right) \left(4 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(2 x -1\right)^{2} \left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 215\)
\(\displaystyle a \! \left(7\right) = 702\)
\(\displaystyle a \! \left(8\right) = 2319\)
\(\displaystyle a \! \left(n +9\right) = \frac{16 \left(1+2 n \right) a \! \left(n \right)}{n +10}-\frac{4 \left(79+48 n \right) a \! \left(1+n \right)}{n +10}+\frac{2 \left(705+277 n \right) a \! \left(n +2\right)}{n +10}-\frac{\left(3333+947 n \right) a \! \left(n +3\right)}{n +10}+\frac{\left(4658+1029 n \right) a \! \left(n +4\right)}{n +10}-\frac{2 \left(2031+365 n \right) a \! \left(n +5\right)}{n +10}+\frac{5 \left(445+67 n \right) a \! \left(n +6\right)}{n +10}-\frac{\left(736+95 n \right) a \! \left(n +7\right)}{n +10}+\frac{\left(133+15 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 9\)
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This specification was found using the strategy pack "Point And Row Placements Tracked Fusion" and has 25 rules.

Found on July 23, 2021.

Finding the specification took 11 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19} \left(x \right)^{2} F_{0}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\ \end{align*}\)