Av(1342, 1423, 2143, 2413)
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Generating Function
\(\displaystyle \frac{\left(x -1+\sqrt{-8 x^{3}+9 x^{2}-6 x +1}\right) \left(x -1\right)}{4 x^{3}-4 x^{2}+2 x}\)
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Counting Sequence
1, 1, 2, 6, 20, 68, 236, 840, 3060, 11360, 42820, 163440, 630452, 2453920, 9626084, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x^{2}-2 x +1\right) F \left(x \right)^{2}-\left(x -1\right)^{2} F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(n +6\right) = -\frac{8 \left(2 n +1\right) a \! \left(n \right)}{7+n}+\frac{2 \left(25 n +42\right) a \! \left(1+n \right)}{7+n}-\frac{2 \left(36 n +97\right) a \! \left(n +2\right)}{7+n}+\frac{\left(61 n +228\right) a \! \left(n +3\right)}{7+n}-\frac{\left(147+31 n \right) a \! \left(n +4\right)}{7+n}+\frac{\left(9 n +52\right) a \! \left(n +5\right)}{7+n}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements" and has 16 rules.

Found on July 23, 2021.

Finding the specification took 2 seconds.

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