Av(1234, 1243, 2314, 2413, 3142)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(-1+x \right)^{3}}{7 x^{4}-14 x^{3}+13 x^{2}-6 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 57, 165, 473, 1358, 3910, 11273, 32509, 93739, 270269, 779222, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(7 x^{4}-14 x^{3}+13 x^{2}-6 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(-1+x \right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +4\right) = -7 a \! \left(n \right)+14 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-\sqrt{2}-1\right) \sqrt{-70+56 \sqrt{2}}-\sqrt{2}+10\right) \left(\frac{1}{2}+\frac{\sqrt{-35+28 \sqrt{2}}}{14}\right)^{-n}}{56}+\\\frac{\left(\left(\mathrm{I} \sqrt{2}-2 \,\mathrm{I}\right) \sqrt{35+28 \sqrt{2}}+\sqrt{2}+10\right) \left(\frac{1}{2}+\frac{\mathrm{I} \sqrt{35+28 \sqrt{2}}}{14}\right)^{-n}}{56}+\\\frac{\left(\left(1+\sqrt{2}\right) \sqrt{-70+56 \sqrt{2}}-\sqrt{2}+10\right) \left(\frac{1}{2}-\frac{\sqrt{-35+28 \sqrt{2}}}{14}\right)^{-n}}{56}-\\\frac{\left(\frac{1}{2}-\frac{\mathrm{I} \sqrt{35+28 \sqrt{2}}}{14}\right)^{-n} \left(\left(\mathrm{I} \sqrt{2}-2 \,\mathrm{I}\right) \sqrt{35+28 \sqrt{2}}-\sqrt{2}-10\right)}{56} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 50 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{13}\! \left(x \right) &= 0\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ \end{align*}\)