Av(1342, 2314, 3412, 4123)
Generating Function
\(\displaystyle -\frac{8 x^{9}-63 x^{8}+186 x^{7}-322 x^{6}+368 x^{5}-289 x^{4}+155 x^{3}-54 x^{2}+11 x -1}{\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(-1+x \right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 20, 63, 181, 481, 1210, 2941, 7015, 16599, 39233, 92989, 221417, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(-1+x \right)^{6} F \! \left(x \right)+8 x^{9}-63 x^{8}+186 x^{7}-322 x^{6}+368 x^{5}-289 x^{4}+155 x^{3}-54 x^{2}+11 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 181\)
\(\displaystyle a \! \left(7\right) = 481\)
\(\displaystyle a \! \left(8\right) = 1210\)
\(\displaystyle a \! \left(9\right) = 2941\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{5}}{120}-\frac{n^{4}}{8}+\frac{3 n^{3}}{8}-\frac{11 n^{2}}{8}-6 a \! \left(n \right)+13 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right)-\frac{53 n}{60}+3, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 181\)
\(\displaystyle a \! \left(7\right) = 481\)
\(\displaystyle a \! \left(8\right) = 1210\)
\(\displaystyle a \! \left(9\right) = 2941\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{5}}{120}-\frac{n^{4}}{8}+\frac{3 n^{3}}{8}-\frac{11 n^{2}}{8}-6 a \! \left(n \right)+13 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right)-\frac{53 n}{60}+3, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \frac{\left(-3410 \left(\left(-\frac{15 \,\mathrm{I}}{31}+\frac{5 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+150040-1240 \,2^{\frac{1}{3}} \left(\left(\frac{3 \,\mathrm{I}}{62}+\frac{\sqrt{3}}{62}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{47 \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(3410 \left(\left(-\frac{15 \,\mathrm{I}}{31}-\frac{5 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+150040+1240 \,2^{\frac{1}{3}} \left(\left(\frac{3 \,\mathrm{I}}{62}-\frac{\sqrt{3}}{62}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(\left(1100 \,2^{\frac{2}{3}} \sqrt{31}\, \sqrt{3}-6820 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+150040+\left(40 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}+2480 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{450120}+\frac{n^{5}}{120}-\frac{n^{4}}{12}-\frac{n^{3}}{24}-\frac{5 n^{2}}{12}-\frac{37 n}{15}+2^{n +1}-2\)
This specification was found using the strategy pack "Point Placements" and has 44 rules.
Found on July 23, 2021.Finding the specification took 9 seconds.
Copy 44 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10} \left(x \right)^{2}\\
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_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= 0\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{13}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{10}\! \left(x \right) F_{33}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{13}\! \left(x \right) F_{33}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{13}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{33}\! \left(x \right)\\
\end{align*}\)