Av(1234, 1243, 2341, 2413, 3142)
Generating Function
$$\displaystyle \frac{\left(x^{3}+2 x -1\right) \left(2 x^{4}-6 x^{3}+9 x^{2}-5 x +1\right)}{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)^{2}}$$
Counting Sequence
1, 1, 2, 6, 19, 55, 150, 396, 1024, 2608, 6565, 16375, 40547, 99809, 244495, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)-\left(x^{3}+2 x -1\right) \left(2 x^{4}-6 x^{3}+9 x^{2}-5 x +1\right) = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(4\right) = 19$$
$$\displaystyle a \! \left(5\right) = 55$$
$$\displaystyle a \! \left(6\right) = 150$$
$$\displaystyle a \! \left(7\right) = 396$$
$$\displaystyle a \! \left(n +7\right) = 2 a \! \left(n \right)-9 a \! \left(n +1\right)+24 a \! \left(n +2\right)-38 a \! \left(n +3\right)+40 a \! \left(n +4\right)-25 a \! \left(n +5\right)+8 a \! \left(n +6\right), \quad n \geq 8$$
Explicit Closed Form
$$\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(552 \,2^{\frac{1}{3}} \left(\left(\left(-\frac{n}{12}-\frac{371}{552}\right) \sqrt{23}+\mathrm{I} n -\frac{151 \,\mathrm{I}}{24}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{4}-\frac{371 \,\mathrm{I}}{184}\right) \sqrt{23}+n -\frac{151}{24}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-345 \left(\left(\left(-\frac{n}{3}+\frac{290}{69}\right) \sqrt{23}+\mathrm{I} n +\frac{100 \,\mathrm{I}}{3}\right) \sqrt{3}+\left(\mathrm{I} n -\frac{290 \,\mathrm{I}}{23}\right) \sqrt{23}-n -\frac{100}{3}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+41400 n +105800\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(345 \,2^{\frac{2}{3}} \left(\left(\left(\frac{n}{3}-\frac{290}{69}\right) \sqrt{23}+\mathrm{I} n +\frac{100 \,\mathrm{I}}{3}\right) \sqrt{3}+\left(\mathrm{I} n -\frac{290 \,\mathrm{I}}{23}\right) \sqrt{23}+n +\frac{100}{3}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-552 \left(\left(\left(\frac{n}{12}+\frac{371}{552}\right) \sqrt{23}+\mathrm{I} n -\frac{151 \,\mathrm{I}}{24}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{4}-\frac{371 \,\mathrm{I}}{184}\right) \sqrt{23}-n +\frac{151}{24}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+41400 n +105800\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(-230 \,2^{\frac{2}{3}} \left(\sqrt{3}\, \left(n -\frac{290}{23}\right) \sqrt{23}+3 n +100\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+92 \left(\sqrt{3}\, \left(n +\frac{371}{46}\right) \sqrt{23}-12 n +\frac{151}{2}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+41400 n +105800\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{317400}\\-2^{n} & \text{otherwise} \end{array}\right.$$

This specification was found using the strategy pack "Point Placements" and has 66 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_{53}\! \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_{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_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \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_{24}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{62}\! \left(x \right)\\ \end{align*}