Av(12345, 12354, 12453, 13452, 21345, 21354, 21453, 23451, 31245, 31254, 31452, 32451, 41235, 41253, 41352, 42351, 51234, 51243, 51342, 52341)
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Generating Function
\(\displaystyle \frac{-6 x^{3} \sqrt{-4 x +1}+2 x^{2} \sqrt{-4 x +1}+16 x^{3}+8 x^{2}-7 x +1}{\left(4 x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 24, 100, 420, 1764, 7392, 30888, 128700, 534820, 2217072, 9170616, 37858184, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x -1\right)^{3} F \left(x \right)^{2}-2 \left(x +1\right) \left(4 x -1\right)^{3} F \! \left(x \right)+36 x^{6}+40 x^{5}+84 x^{4}-20 x^{3}-25 x^{2}+10 x -1 = 0\)
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Recurrence
\(\displaystyle a{\left(0 \right)} = 1\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(n + 2 \right)} = - \frac{6 \left(2 n - 3\right) a{\left(n \right)}}{n} + \frac{\left(7 n - 4\right) a{\left(n + 1 \right)}}{n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 8 rules.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{6}\! \left(x , y\right) &= y^{2} x^{2}+4 x F_{6}\! \left(x , y\right)^{2} y -8 x F_{6}\! \left(x , y\right) y +4 y x -F_{6}\! \left(x , y\right)^{2}+3 F_{6}\! \left(x , y\right)-1\\ F_{7}\! \left(x \right) &= x\\ \end{align*}\)