Av(1324, 1342, 2431, 4132)
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Generating Function
\(\displaystyle \frac{\left(-2 x^{4}+4 x^{3}-6 x^{2}+4 x -1\right) \sqrt{1-4 x}-2 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{2 x^{2} \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 20, 66, 219, 738, 2529, 8796, 30975, 110210, 395553, 1430256, 5204968, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x -1\right)^{4} F \left(x \right)^{2}+\left(2 x^{5}-6 x^{4}+14 x^{3}-14 x^{2}+6 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-2 x^{7}+6 x^{6}-12 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 219\)
\(\displaystyle a \! \left(7\right) = 738\)
\(\displaystyle a \! \left(8\right) = 2529\)
\(\displaystyle a \! \left(n +6\right) = -\frac{4 \left(-1+2 n \right) a \! \left(n \right)}{8+n}-\frac{2 \left(52+23 n \right) a \! \left(2+n \right)}{8+n}+\frac{2 \left(7+13 n \right) a \! \left(n +1\right)}{8+n}+\frac{2 \left(93+25 n \right) a \! \left(n +3\right)}{8+n}-\frac{2 \left(77+15 n \right) a \! \left(n +4\right)}{8+n}+\frac{\left(59+9 n \right) a \! \left(n +5\right)}{8+n}+\frac{2}{8+n}, \quad n \geq 9\)
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This specification was found using the strategy pack "Point Placements" and has 28 rules.

Found on July 23, 2021.

Finding the specification took 9 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_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \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_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{5}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{7}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{26}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{26}\! \left(x \right) F_{9}\! \left(x \right)\\ \end{align*}\)