Av(1234, 1243, 2314, 2341, 3214)
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Generating Function
\(\displaystyle -\frac{4 x^{4}+2 x^{3}+x^{2}-3 x +1}{\left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 154, 422, 1139, 3045, 8088, 21388, 56387, 148345, 389700, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+4 x^{4}+2 x^{3}+x^{2}-3 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{\left(\left(\left(\left(-\frac{7 \sqrt{3}}{132}-\frac{29 \,\mathrm{I}}{132}\right) \sqrt{11}+\frac{\mathrm{I} \sqrt{3}}{4}+\frac{1}{4}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\frac{5 \sqrt{3}}{132}-\frac{49 \,\mathrm{I}}{132}\right) \sqrt{11}+\frac{3 \,\mathrm{I} \sqrt{3}}{4}-\frac{1}{4}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{3}{2}+\frac{\mathrm{I} \sqrt{11}}{66}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}+\left(\left(-\frac{1}{2}+\left(\frac{3 \sqrt{3}}{22}-\frac{\mathrm{I}}{33}\right) \sqrt{11}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-\frac{9 \sqrt{3}}{44}-\frac{17 \,\mathrm{I}}{132}\right) \sqrt{11}+\frac{\mathrm{I} \sqrt{3}}{4}+\frac{5}{4}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{3}{2}-\frac{\mathrm{I} \sqrt{11}}{66}\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}+\frac{\left(\sqrt{5}-5\right) \left(\left(\left(-\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{11}+\mathrm{I} \sqrt{3}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+16+\left(\left(-2 \,\mathrm{I}-\frac{2 \sqrt{3}}{3}\right) \sqrt{11}+4 \,\mathrm{I} \sqrt{3}+4\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}-\frac{\left(5+\sqrt{5}\right) \left(\left(\left(-\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{11}+\mathrm{I} \sqrt{3}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+16+\left(\left(-2 \,\mathrm{I}-\frac{2 \sqrt{3}}{3}\right) \sqrt{11}+4 \,\mathrm{I} \sqrt{3}+4\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}\right) \left(\left(\left(-\frac{27 \,\mathrm{I}}{22}-\frac{9 \sqrt{3}}{22}\right) \sqrt{11}+\frac{5 \,\mathrm{I} \sqrt{3}}{2}+\frac{5}{2}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-8+\left(\left(-\frac{9 \,\mathrm{I}}{11}+\frac{3 \sqrt{3}}{11}\right) \sqrt{11}+\mathrm{I} \sqrt{3}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right)}{24}\)
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This specification was found using the strategy pack "Point Placements" and has 43 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_{18}\! \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_{29}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \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_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{20}\! \left(x \right)\\ \end{align*}\)