Av(1342, 2314, 2413, 3412)
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Generating Function
\(\displaystyle \frac{\left(x^{3}-5 x^{2}+4 x -1\right) \sqrt{1-4 x}-6 x^{4}+5 x^{3}+3 x^{2}-4 x +1}{2 x \left(2 x -1\right) \left(-1+x \right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 20, 67, 222, 732, 2423, 8102, 27452, 94321, 328391, 1157012, 4118998, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(-1+x \right)^{6} F \left(x \right)^{2}+\left(3 x^{3}-x^{2}-2 x +1\right) \left(2 x -1\right)^{2} \left(-1+x \right)^{3} F \! \left(x \right)+9 x^{7}-14 x^{6}-13 x^{5}+55 x^{4}-61 x^{3}+33 x^{2}-9 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 222\)
\(\displaystyle a \! \left(7\right) = 732\)
\(\displaystyle a \! \left(n +6\right) = -\frac{4 \left(3+2 n \right) a \! \left(n \right)}{7+n}+\frac{2 \left(74+27 n \right) a \! \left(1+n \right)}{7+n}-\frac{\left(359+109 n \right) a \! \left(n +2\right)}{7+n}+\frac{2 \left(201+50 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(231+47 n \right) a \! \left(n +4\right)}{7+n}+\frac{\left(65+11 n \right) a \! \left(n +5\right)}{7+n}+\frac{-11+5 n}{7+n}, \quad n \geq 8\)
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This specification was found using the strategy pack "Point Placements" and has 29 rules.

Found on July 23, 2021.

Finding the specification took 10 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_{10}\! \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_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{10}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{23}\! \left(x \right) &= 0\\ F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ \end{align*}\)