Av(1243, 1432, 2431, 4231)
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Generating Function
\(\displaystyle -\frac{3 x^{6}-19 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 200, 586, 1656, 4565, 12378, 33200, 88410, 234297, 618862, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+3 x^{6}-19 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 200\)
\(\displaystyle a \! \left(n +3\right) = -\frac{n^{3}}{6}+\frac{n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{2 n}{3}+1, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle -\frac{n^{3}}{6}+\frac{n^{2}}{2}-\frac{4 n}{3}+2-\frac{2 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{2 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-2^{n}\)
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This specification was found using the strategy pack "Row And Col Placements" and has 59 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_{20}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{20}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{46}\! \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_{20}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= x\\ F_{21}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{23}\! \left(x \right) &= 0\\ F_{24}\! \left(x \right) &= F_{18}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{20}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{20}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{20}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{20}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{20}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{20}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{20}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{43}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{20}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{20}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{20}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{20}\! \left(x \right) F_{49}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{20}\! \left(x \right) F_{52}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{20}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{17} \left(x \right)^{2} F_{20}\! \left(x \right) F_{29}\! \left(x \right)\\ \end{align*}\)