Av(1243, 1342, 4132)
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Generating Function
\(\displaystyle \frac{\left(-x^{3}+4 x^{2}-4 x +1\right) \sqrt{1-4 x}-4 x^{4}+15 x^{3}-20 x^{2}+8 x -1}{2 x \left(x^{3}-4 x^{2}+5 x -1\right) \left(x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 21, 77, 286, 1067, 3992, 14976, 56338, 212517, 803758, 3047409, 11580777, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{3}-4 x^{2}+5 x -1\right) \left(x -1\right)^{2} F \left(x \right)^{2}+\left(x -1\right) \left(4 x^{4}-15 x^{3}+20 x^{2}-8 x +1\right) F \! \left(x \right)+4 x^{4}-13 x^{3}+16 x^{2}-7 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 286\)
\(\displaystyle a \! \left(7\right) = 1067\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(69+29 n \right) a \! \left(1+n \right)}{7+n}-\frac{\left(247+79 n \right) a \! \left(n +2\right)}{7+n}+\frac{2 \left(193+49 n \right) a \! \left(n +3\right)}{7+n}-\frac{2 \left(129+26 n \right) a \! \left(n +4\right)}{7+n}+\frac{12 \left(n +6\right) a \! \left(n +5\right)}{7+n}-\frac{3 \left(n +3\right)}{7+n}, \quad n \geq 8\)
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This specification was found using the strategy pack "Point And Row Placements" and has 27 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_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{11}\! \left(x \right) &= x\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{19} \left(x \right)^{2} F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{19} \left(x \right)^{2} F_{11}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{15}\! \left(x \right) F_{19}\! \left(x \right) F_{3}\! \left(x \right)\\ \end{align*}\)