Av(12354, 12453, 12543, 13254, 21354, 21453, 21543, 23154, 31254, 31452, 31542, 32154, 41253, 41352, 41532, 42153, 51243, 51342, 51432, 52143)
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Counting Sequence
1, 1, 2, 6, 24, 100, 426, 1848, 8120, 36018, 160940, 723338, 3266496, 14809366, 67365298, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(-4 x^{4}+12 x^{3}-4 x^{2}+24 x -5\right) F \left(x \right)^{3}+\left(x +3\right) \left(4 x^{4}-12 x^{3}+4 x^{2}-24 x +5\right) F \left(x \right)^{2}+\left(-8 x^{5}+12 x^{4}+36 x^{3}+37 x^{2}+62 x -15\right) F \! \left(x \right)+\left(4 x^{2}+4 x -1\right) \left(x^{3}-3 x^{2}-x -5\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 100\)
\(\displaystyle a \! \left(6\right) = 426\)
\(\displaystyle a \! \left(7\right) = 1848\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(n -1\right) \left(n +1\right) a \! \left(n \right)}{5 \left(n +6\right) \left(n +5\right)}-\frac{3 n \left(7 n +11\right) a \! \left(n +1\right)}{5 \left(n +6\right) \left(n +5\right)}+\frac{7 \left(14 n +45\right) \left(n +1\right) a \! \left(n +2\right)}{10 \left(n +6\right) \left(n +5\right)}-\frac{\left(91 n^{2}+459 n +590\right) a \! \left(n +3\right)}{5 \left(n +6\right) \left(n +5\right)}+\frac{\left(89 n^{2}+493 n +660\right) a \! \left(n +4\right)}{5 \left(n +6\right) \left(n +5\right)}-\frac{\left(493 n^{2}+3625 n +6414\right) a \! \left(n +5\right)}{20 \left(n +6\right) \left(n +5\right)}+\frac{3 \left(31 n +133\right) a \! \left(n +6\right)}{10 \left(n +6\right)}, \quad n \geq 8\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 23 rules.

Found on January 23, 2022.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= \frac{y F_{7}\! \left(x , y\right)-F_{7}\! \left(x , 1\right)}{-1+y}\\ F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= y x\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ \end{align*}\)