Av(1243, 1342, 4132, 4213)
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Generating Function
\(\displaystyle -\frac{x^{5}-5 x^{4}+7 x^{3}-9 x^{2}+5 x -1}{\left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 193, 559, 1583, 4431, 12332, 34228, 94882, 262870, 728099, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{5}-5 x^{4}+7 x^{3}-9 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+n \left(n +2\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \frac{\left(\left(\left(-52 \sqrt{3}-156 \,\mathrm{I}\right) \sqrt{19}+380 \,\mathrm{I} \sqrt{3}+380\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+2432+\left(\left(51 \,\mathrm{I}-17 \sqrt{3}\right) \sqrt{19}+133 \,\mathrm{I} \sqrt{3}-133\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{14592}+\frac{\left(\left(\left(156 \,\mathrm{I}-52 \sqrt{3}\right) \sqrt{19}-380 \,\mathrm{I} \sqrt{3}+380\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+2432+\left(\left(-51 \,\mathrm{I}-17 \sqrt{3}\right) \sqrt{19}-133 \,\mathrm{I} \sqrt{3}-133\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{14592}+\frac{\left(\left(104 \sqrt{19}\, \sqrt{3}-760\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+34 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}+266 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}+2432\right) \left(\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{1}{3}+\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}}{64}\right)^{-n}}{14592}-\frac{n^{2}}{2}+\frac{1}{2}\)
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This specification was found using the strategy pack "Point Placements" and has 110 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{25}\! \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_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{33}\! \left(x \right) &= 0\\ 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_{55}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{33}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{47}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{57}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{60}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{33}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{57}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{75}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{76}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{83}\! \left(x \right) &= 3 F_{33}\! \left(x \right)+F_{84}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{95}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{100}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{101}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{105}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{100}\! \left(x \right)\\ \end{align*}\)