Av(1243, 1342, 3124, 4123)
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Generating Function
\(\displaystyle \frac{\left(-x^{4}+8 x^{3}-10 x^{2}+5 x -1\right) \sqrt{1-4 x}+2 x^{5}+3 x^{4}-10 x^{3}+10 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 207, 660, 2135, 7042, 23687, 81097, 281902, 992556, 3532535, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \left(x \right)^{2}-\left(x^{4}+2 x^{3}-4 x^{2}+3 x -1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{9}+4 x^{8}-24 x^{7}+83 x^{6}-156 x^{5}+168 x^{4}-110 x^{3}+44 x^{2}-10 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 207\)
\(\displaystyle a \! \left(7\right) = 660\)
\(\displaystyle a \! \left(8\right) = 2135\)
\(\displaystyle a \! \left(9\right) = 7042\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(77+39 n \right) a \! \left(n +1\right)}{n +8}+\frac{\left(532+199 n \right) a \! \left(n +2\right)}{n +8}-\frac{3 \left(280+79 n \right) a \! \left(n +3\right)}{n +8}+\frac{6 \left(119+26 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(336+59 n \right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(41+6 n \right) a \! \left(n +6\right)}{n +8}+\frac{2 n -6}{n +8}, \quad n \geq 10\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 47 rules.

Found on July 23, 2021.

Finding the specification took 5 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_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{32}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{11}\! \left(x \right) F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \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_{12}\! \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_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \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_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x , 1\right)\\ F_{27}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= \frac{y F_{29}\! \left(x , y\right)-F_{29}\! \left(x , 1\right)}{-1+y}\\ F_{29}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= y x\\ F_{32}\! \left(x \right) &= F_{11}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{11}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x , 1\right)\\ F_{39}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{31}\! \left(x , y\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 , 1\right)\\ F_{42}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)+F_{39}\! \left(x , y\right)+F_{43}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{44}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= \frac{y F_{28}\! \left(x , y\right)-F_{28}\! \left(x , 1\right)}{-1+y}\\ F_{45}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{46}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= \frac{y F_{42}\! \left(x , y\right)-F_{42}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)