Av(1243, 1342, 3412, 4132, 4231)
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Generating Function
\(\displaystyle \frac{x^{5}-4 x^{4}+8 x^{3}-10 x^{2}+5 x -1}{\left(x -1\right)^{4} \left(2 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 54, 138, 326, 729, 1570, 3296, 6802, 13879, 28110, 56662, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{4} \left(2 x -1\right) F \! \left(x \right)-x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{n \left(n +4\right) \left(-1+n \right)}{6}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -3+7 \,2^{-1+n}-\frac{n^{3}}{6}-n^{2}-\frac{11 n}{6} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 60 rules.

Found on July 23, 2021.

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_{14}\! \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_{14}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11} \left(x \right)^{3}\\ 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_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{14}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{14}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{14}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{14}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{14}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{14}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{14}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{57}\! \left(x \right)\\ \end{align*}\)