Av(1234, 2143, 2413)
Generating Function
\(\displaystyle \frac{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}{\left(2 x^{3}-3 x^{2}+4 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 247, 821, 2704, 8868, 29030, 94960, 310531, 1015359, 3319829, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{3}-3 x^{2}+4 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-\left(x^{3}-2 x^{2}+3 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 247\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+\frac{n \left(n +1\right)}{2}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 247\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+\frac{n \left(n +1\right)}{2}, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(165 \,\mathrm{I} \,3^{\frac{1}{3}}-55 \,3^{\frac{5}{6}}\right) \sqrt{38}-285 \,\mathrm{I} \,3^{\frac{5}{6}}+285 \,3^{\frac{1}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}+5700+\left(\left(-3 \,\mathrm{I} \,3^{\frac{2}{3}}-3 \,3^{\frac{1}{6}}\right) \sqrt{38}-399 \,\mathrm{I} \,3^{\frac{1}{6}}-133 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(27+6 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \,3^{\frac{5}{6}} \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{2}+\frac{\left(\left(9 \,\mathrm{I}-2 \sqrt{38}\right) 3^{\frac{1}{6}}+\left(-2 \,\mathrm{I} \sqrt{38}+3\right) 3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}\right)^{-n}}{68400}\\+\\\frac{\left(\left(\left(-165 \,\mathrm{I} \,3^{\frac{1}{3}}-55 \,3^{\frac{5}{6}}\right) \sqrt{38}+285 \,\mathrm{I} \,3^{\frac{5}{6}}+285 \,3^{\frac{1}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}+5700+\left(\left(3 \,\mathrm{I} \,3^{\frac{2}{3}}-3 \,3^{\frac{1}{6}}\right) \sqrt{38}+399 \,\mathrm{I} \,3^{\frac{1}{6}}-133 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(27+6 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \,3^{\frac{5}{6}} \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{2}+\frac{\left(\left(-9 \,\mathrm{I}-2 \sqrt{38}\right) 3^{\frac{1}{6}}+\left(2 \,\mathrm{I} \sqrt{38}+3\right) 3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}\right)^{-n}}{68400}\\+\\\frac{\left(\left(110 \,3^{\frac{5}{6}} \sqrt{38}-570 \,3^{\frac{1}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}+5700+\left(6 \sqrt{38}\, 3^{\frac{1}{6}}+266 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(2 \sqrt{38}\, 3^{\frac{1}{6}}-3 \,3^{\frac{2}{3}}\right) \left(9+2 \sqrt{38}\, \sqrt{3}\right)^{\frac{2}{3}}}{150}-\frac{\left(27+6 \sqrt{38}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{2}\right)^{-n}}{68400}\\-\frac{n^{2}}{4}+\frac{n}{4}+\frac{1}{4} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 119 rules.
Found on January 18, 2022.Finding the specification took 9 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 119 equations to clipboard:
\(\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_{29}\! \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_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{4}\! \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_{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_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{15}\! \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_{40}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \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_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{94}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{79}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{86}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{111}\! \left(x \right)+F_{15}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{109}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{103}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{118}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{79}\! \left(x \right)\\
\end{align*}\)