Av(1324, 2413, 3241)
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Generating Function
\(\displaystyle \frac{\left(-x^{2}+5 x -2\right) \sqrt{1-4 x}+3 x^{2}-7 x +2}{2 \left(x^{2}-3 x +1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 21, 76, 275, 989, 3539, 12631, 45066, 161021, 576887, 2074166, 7488003, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right)^{2} x F \left(x \right)^{2}-\left(x -2\right) \left(3 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{4}-8 x^{3}+21 x^{2}-12 x +2 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = \frac{\left(1+2 n \right) a \! \left(n \right)}{n +6}-\frac{\left(58+33 n \right) a \! \left(1+n \right)}{2 \left(n +6\right)}+\frac{\left(107+40 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(116+31 n \right) a \! \left(n +3\right)}{n +6}+\frac{\left(92+19 n \right) a \! \left(n +4\right)}{2 n +12}, \quad n \geq 5\)
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This specification was found using the strategy pack "Point Placements" and has 21 rules.

Found on July 23, 2021.

Finding the specification took 4 seconds.

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