Av(1234, 1342, 1423, 2341, 4123)
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Generating Function
\(\displaystyle \frac{\left(x -1\right) \sqrt{-4 x +1}-2 x^{2}-x +1}{2 x \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 19, 60, 191, 619, 2048, 6909, 23704, 82489, 290500, 1033399, 3707838, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x \right)^{2}+\left(x +1\right) \left(2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{3}+2 x^{2}-3 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +2\right) = -\frac{2 \left(2 n +3\right) a \! \left(n \right)}{3+n}+\frac{\left(5 n +9\right) a \! \left(1+n \right)}{3+n}+\frac{3 n +3}{3+n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Requirement Placements Tracked Fusion" and has 53 rules.

Found on January 18, 2022.

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