Av(1342, 1432, 2431, 4132)
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Generating Function
\(\displaystyle \frac{\left(-x^{3}+2 x^{2}-3 x +1\right) \sqrt{1-4 x}+5 x^{3}-8 x^{2}+5 x -1}{4 x^{3}-2 x^{2}}\)
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Counting Sequence
1, 1, 2, 6, 20, 69, 241, 847, 2992, 10625, 37942, 136268, 492176, 1787330, 6523973, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(2 x -1\right)^{2} F \left(x \right)^{2}-\left(2 x -1\right) \left(5 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)+x^{5}+2 x^{4}-9 x^{3}+12 x^{2}-6 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 69\)
\(\displaystyle a \! \left(n +5\right) = \frac{4 \left(-1+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(109+37 n \right) a \! \left(2+n \right)}{7+n}-\frac{2 \left(12+11 n \right) a \! \left(n +1\right)}{7+n}-\frac{2 \left(61+14 n \right) a \! \left(n +3\right)}{7+n}+\frac{3 \left(17+3 n \right) a \! \left(n +4\right)}{7+n}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements" and has 19 rules.

Found on July 23, 2021.

Finding the specification took 12 seconds.

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