Av(1234, 1342, 1423, 2314, 3124)
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Generating Function
\(\displaystyle -\frac{\left(-3 x^{3}+3 x^{2}+\left(x -1\right)^{3} \sqrt{-4 x +1}-3 x +1\right) \left(x -1\right)^{3}}{2 x \left(x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 59, 184, 587, 1921, 6426, 21886, 75667, 264956, 937922, 3351336, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1\right) F \left(x \right)^{2}-\left(3 x^{3}-3 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x -1\right)^{6} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 184\)
\(\displaystyle a \! \left(7\right) = 587\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}+\frac{\left(40+21 n \right) a \! \left(1+n \right)}{9+n}-\frac{23 \left(7+3 n \right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(227+66 n \right) a \! \left(n +3\right)}{9+n}-\frac{17 \left(40+9 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(613+111 n \right) a \! \left(n +5\right)}{9+n}-\frac{16 \left(20+3 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(86+11 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
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This specification was found using the strategy pack "Insertion Point Placements Tracked Fusion" and has 55 rules.

Found on July 23, 2021.

Finding the specification took 9 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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x , 1\right)\\ F_{23}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= y x\\ F_{33}\! \left(x , y\right) &= F_{34}\! \left(x \right)+F_{36}\! \left(x , y\right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{43}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{13}\! \left(x \right)+F_{40}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= -\frac{y \left(F_{24}\! \left(x , 1\right)-F_{24}\! \left(x , y\right)\right)}{-1+y}\\ F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{13} \left(x \right)^{2} F_{30}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= -\frac{y \left(F_{36}\! \left(x , 1\right)-F_{36}\! \left(x , y\right)\right)}{-1+y}\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{13} \left(x \right)^{2}\\ F_{50}\! \left(x \right) &= F_{33}\! \left(x , 1\right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\ \end{align*}\)