Av(1243, 1342, 2314, 3412)
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Generating Function
\(\displaystyle -\frac{8 x^{8}-41 x^{7}+111 x^{6}-174 x^{5}+172 x^{4}-110 x^{3}+44 x^{2}-10 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{7}}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 189, 516, 1320, 3211, 7524, 17158, 38376, 84654, 184883, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x -1\right)^{7} F \! \left(x \right)+8 x^{8}-41 x^{7}+111 x^{6}-174 x^{5}+172 x^{4}-110 x^{3}+44 x^{2}-10 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 189\)
\(\displaystyle a \! \left(7\right) = 516\)
\(\displaystyle a \! \left(8\right) = 1320\)
\(\displaystyle a \! \left(n +2\right) = \frac{n^{6}}{720}-\frac{7 n^{5}}{240}+\frac{23 n^{4}}{144}-\frac{17 n^{3}}{48}+\frac{241 n^{2}}{180}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{67 n}{60}+2, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle 4+\frac{7 n}{10}-3 \,2^{n}-\frac{3 n^{3}}{16}+\frac{5 n^{4}}{144}-\frac{n^{5}}{80}+\frac{n^{6}}{720}+2^{n} n +\frac{167 n^{2}}{360}\)
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This specification was found using the strategy pack "Row And Col Placements" and has 50 rules.

Found on July 23, 2021.

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_{11}\! \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_{11}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{39}\! \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_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= x\\ F_{12}\! \left(x \right) &= F_{1}\! \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) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{11}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{36}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{9}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{11}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{11}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{11}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{11}\! \left(x \right) F_{45}\! \left(x \right)\\ \end{align*}\)