Av(1243, 1342, 1432, 2314, 3142)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x -1\right)^{4}}{8 x^{5}-21 x^{4}+27 x^{3}-19 x^{2}+7 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 58, 173, 512, 1512, 4461, 13154, 38775, 114290, 336878, 993015, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(8 x^{5}-21 x^{4}+27 x^{3}-19 x^{2}+7 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(n +5\right) = 8 a \! \left(n \right)-21 a \! \left(n +1\right)+27 a \! \left(n +2\right)-19 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \frac{3056 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{4757}+\frac{3056 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{4757}+\frac{3056 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +3}}{4757}+\frac{3056 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +3}}{4757}+\frac{3056 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +3}}{4757}-\frac{9054 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{4757}-\frac{9054 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{4757}-\frac{9054 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{4757}-\frac{9054 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +2}}{4757}-\frac{9054 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +2}}{4757}+\frac{10333 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{4757}+\frac{10333 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{4757}+\frac{10333 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{4757}+\frac{10333 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{4757}+\frac{10333 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +1}}{4757}+\frac{1487 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n -1}}{4757}+\frac{1487 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n -1}}{4757}+\frac{1487 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n -1}}{4757}+\frac{1487 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n -1}}{4757}+\frac{1487 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n -1}}{4757}-\frac{5704 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{4757}-\frac{5704 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{4757}-\frac{5704 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{4757}-\frac{5704 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{4757}-\frac{5704 \mathit{RootOf} \left(8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{4757}+\left(\left\{\begin{array}{cc}\frac{1}{4} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Point Placements" and has 61 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{18}\! \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_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{56}\! \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_{50}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{12}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\ \end{align*}\)