Av(1324, 1342, 1423, 1432, 2413, 2431, 4132)
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Generating Function
\(\displaystyle \frac{\left(x^{2}-x -1\right) \sqrt{1-4 x}-3 x^{2}+x +1}{2 x}\)
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Counting Sequence
1, 1, 2, 6, 17, 51, 160, 519, 1727, 5863, 20228, 70720, 250002, 892126, 3209328, ...
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Implicit Equation for the Generating Function
\(\displaystyle x F \left(x \right)^{2}+\left(3 x^{2}-x -1\right) F \! \left(x \right)+x^{4}-2 x^{2}+x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 17\)
\(\displaystyle a \! \left(n +3\right) = -\frac{2 \left(-3+2 n \right) a \! \left(n \right)}{4+n}+\frac{\left(2+5 n \right) a \! \left(1+n \right)}{4+n}+\frac{\left(8+3 n \right) a \! \left(n +2\right)}{4+n}, \quad n \geq 5\)
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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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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{5} \left(x \right)^{3} F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ \end{align*}\)