Av(1234, 1324, 1342, 2431, 3124)
Generating Function
\(\displaystyle -\frac{2 x^{6}-7 x^{5}+15 x^{4}-19 x^{3}+15 x^{2}-6 x +1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 153, 395, 981, 2375, 5657, 13339, 31262, 73006, 170142, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+2 x^{6}-7 x^{5}+15 x^{4}-19 x^{3}+15 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 153\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{3}}{6}+\frac{n^{2}}{2}+a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{7 n}{3}+1, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 153\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{3}}{6}+\frac{n^{2}}{2}+a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{7 n}{3}+1, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{\left(-115 \,2^{\frac{2}{3}} \left(\left(-\frac{27 \,\mathrm{I}}{23}+\frac{9 \sqrt{3}}{23}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2300-161 \,2^{\frac{1}{3}} \left(\left(-\frac{9 \,\mathrm{I}}{161}-\frac{3 \sqrt{3}}{161}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{6900}+\frac{\left(115 \,2^{\frac{2}{3}} \left(\left(-\frac{27 \,\mathrm{I}}{23}-\frac{9 \sqrt{3}}{23}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2300+161 \,2^{\frac{1}{3}} \left(\left(-\frac{9 \,\mathrm{I}}{161}+\frac{3 \sqrt{3}}{161}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{6900}+\frac{\left(\left(90 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{23}-230 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2300+\left(-6 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+322 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{6900}-\frac{n^{3}}{6}-\frac{11 n}{6}\)
This specification was found using the strategy pack "Point Placements" and has 60 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 60 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_{14}\! \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_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \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_{20}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{40}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{55}\! \left(x \right)\\
\end{align*}\)