Av(1342, 2341, 3412)
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Generating Function
\(\displaystyle -\frac{\left(6 \sqrt{1-4 x}\, x^{3}+18 x^{4}-9 \sqrt{1-4 x}\, x^{2}-38 x^{3}+5 \sqrt{1-4 x}\, x +29 x^{2}-\sqrt{1-4 x}-9 x +1\right) \left(2 x -1\right)}{2 x \left(9 x^{3}-14 x^{2}+7 x -1\right) \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 21, 77, 285, 1055, 3905, 14476, 53812, 200709, 751206, 2820944, 10625962, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(9 x^{3}-14 x^{2}+7 x -1\right) \left(x -1\right)^{4} F \left(x \right)^{2}+\left(2 x -1\right) \left(18 x^{4}-38 x^{3}+29 x^{2}-9 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(9 x^{4}-20 x^{3}+18 x^{2}-7 x +1\right) \left(2 x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 285\)
\(\displaystyle a \! \left(7\right) = 1055\)
\(\displaystyle a \! \left(8\right) = 3905\)
\(\displaystyle a \! \left(n +8\right) = -\frac{108 \left(2 n +3\right) a \! \left(n \right)}{n +9}+\frac{6 \left(155 n +336\right) a \! \left(1+n \right)}{n +9}-\frac{3 \left(577 n +1720\right) a \! \left(n +2\right)}{n +9}+\frac{\left(7105+1822 n \right) a \! \left(n +3\right)}{n +9}-\frac{5 \left(237 n +1159\right) a \! \left(n +4\right)}{n +9}+\frac{2 \left(243 n +1439\right) a \! \left(n +5\right)}{n +9}-\frac{2 \left(61 n +425\right) a \! \left(n +6\right)}{n +9}+\frac{17 \left(n +8\right) a \! \left(n +7\right)}{n +9}+\frac{2}{n +9}, \quad n \geq 9\)
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This specification was found using the strategy pack "Insertion Point Col Placements" and has 53 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_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{6} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{2}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{37}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{2}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{14}\! \left(x \right) F_{47}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{14}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ \end{align*}\)