Av(1243, 1324, 1342, 3124, 4123)
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Generating Function
\(\displaystyle \frac{\left(-x^{4}-3 x^{3}+6 x^{2}-4 x +1\right) \sqrt{1-4 x}-2 x^{5}-x^{4}+5 x^{3}-6 x^{2}+4 x -1}{4 x \left(x -\frac{1}{2}\right) \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 19, 59, 184, 585, 1904, 6337, 21507, 74206, 259591, 918666, 3282879, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x \right)^{2}+\left(x^{4}+x^{3}-2 x^{2}+2 x -1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+\left(x^{3}+2 x^{2}-3 x +1\right) \left(x^{6}+4 x^{4}-10 x^{3}+10 x^{2}-5 x +1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 184\)
\(\displaystyle a \! \left(7\right) = 585\)
\(\displaystyle a \! \left(8\right) = 1904\)
\(\displaystyle a \! \left(9\right) = 6337\)
\(\displaystyle a \! \left(n +7\right) = -\frac{4 \left(2 n -1\right) a \! \left(n \right)}{n +8}-\frac{2 \left(5 n +24\right) a \! \left(1+n \right)}{n +8}+\frac{\left(221+83 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(136 n +467\right) a \! \left(n +3\right)}{n +8}+\frac{\left(492+109 n \right) a \! \left(n +4\right)}{n +8}-\frac{16 \left(3 n +17\right) a \! \left(n +5\right)}{n +8}+\frac{\left(75+11 n \right) a \! \left(n +6\right)}{n +8}+\frac{2}{n +8}, \quad n \geq 10\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 44 rules.

Found on January 20, 2022.

Finding the specification took 5 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_{29}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{23}\! \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_{10}\! \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_{14}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\ F_{24}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= y x\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{28}\! \left(x , y\right) &= \frac{y F_{24}\! \left(x , y\right)-F_{24}\! \left(x , 1\right)}{-1+y}\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{28}\! \left(x , 1\right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x , 1\right)\\ F_{37}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \left(x , y\right)+F_{41}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{39}\! \left(x , y\right) &= \frac{y F_{40}\! \left(x , y\right)-F_{40}\! \left(x , 1\right)}{-1+y}\\ F_{40}\! \left(x , y\right) &= \frac{y F_{24}\! \left(x , y\right)-F_{24}\! \left(x , 1\right)}{-1+y}\\ F_{41}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{43}\! \left(x , y\right) &= \frac{y F_{37}\! \left(x , y\right)-F_{37}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)