Av(1324, 1342, 1423, 1432, 2413, 2431, 3142, 4132)
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Generating Function
\(\displaystyle \frac{-\sqrt{-4 x +1}\, x^{2}-2 x^{3}+x^{2}-\sqrt{-4 x +1}+1}{2 x}\)
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Counting Sequence
1, 1, 2, 6, 16, 47, 146, 471, 1562, 5291, 18226, 63648, 224808, 801686, 2882452, ...
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Implicit Equation for the Generating Function
\(\displaystyle x F \left(x \right)^{2}+\left(x -1\right) \left(2 x^{2}+x +1\right) F \! \left(x \right)+x^{5}+x^{2}+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 16\)
\(\displaystyle a \! \left(5\right) = 47\)
\(\displaystyle a \! \left(n +3\right) = \frac{2 \left(-3+2 n \right) a \! \left(n \right)}{n +4}-\frac{n a \! \left(1+n \right)}{n +4}+\frac{2 \left(5+2 n \right) a \! \left(n +2\right)}{n +4}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements" and has 12 rules.

Found on July 23, 2021.

Finding the specification took 1 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{8}\! \left(x \right)\\ \end{align*}\)